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Here is an illustration of Method 1.

Example. Let $Q(x) = x^2 - 2$ in $\mathbf F_5[x]$. Then $\mathbf F_5[x]/(Q^3) \cong \mathbf F_{25}[t]/(t^3)$. To write down an explicit ring isomorphism we need a copy of $\mathbf F_{25}$ in $\mathbf F_5[x]/(Q^3)$. We can view $\mathbf F_{25}$ as $\mathbf F_5(\alpha)$ where $\alpha^2 = 2$. One root of $t^2 - 2$ in $\mathbf F_5[x]/(x^2-2)$ is $x \bmod x^2-2$, and the unique lifting of this to a root of $t^2 - 2$ in $\mathbf F_5[x]/(Q^3)$ is $$ r = x + xQ + 4xQ^2 = x + x(x^2-2) + 4x(x^2-2)^2 \bmod Q^3 $$ by an explicit search. Therefore the copy of $\mathbf F_{25}$ inside $\mathbf F_5[x]/(Q^3)$ is $\mathbf F = \mathbf F_5 + \mathbf F_5r$ and $$ \mathbf F_5[x]/(Q^3) = \mathbf F + \mathbf F Q + \mathbf F Q^2 \bmod Q^3. $$ The right side naturally looks like $\mathbf F_{25}[t]/(t^3)$ and we get an isomorphism $\mathbf F_{25}[t]/(t^3) \to \mathbf F_5[x]/(Q^3)$ once we decide on an isomorphism between $\mathbf F_{25}$ and $\mathbf F$.

The main idea is to view both rings as quotient rings of completions: $\mathcal O_K/\mathfrak p^e \cong \widehat{\mathcal O_{\mathfrak p}}/\widehat{\mathfrak p}^e$ and $k[t]/(t^e) \cong k[[t]]/(t^e)$. Then the structure of the ring of integers of local fields, in characteristic $0$ and characteristic $p$, will help us solve the problem.

The last step is to show $\mathbf F_p[x]/(Q(x)^e) \cong k[t]/(t^e)$ as rings. Since $Q(x)$ is irreducible in $\mathbf F_p[x]$, the ring $\mathbf F_p[x]/(Q(x)^e)$ is unchanged up to isomorphism if we replace $\mathbf F_p[x]$ with its $Q$-adic completion $\mathbf F_p[x]_Q$: $$ \mathbf F_p[x]/(Q(x)^e) \cong \mathbf F_p[x]_Q/(Q(x)^e) $$ as rings. The residue field of $\mathbf F_p[x]_Q$ is isomorphic to $\mathbf F_p[x]/(Q(x))$, which is isomorphic to $k$. The completion $\mathbf F_p[x]_Q$ is the ring of integers of the $Q$-adic field completion $\mathbf F_p(x)_Q$, and the structure of local fields of positive characteristic says they are all isomorphic to the formal Laurent series field over the residue field. Thus $\mathbf F_p(x)_Q \cong k((t))$ as valued fields, so their rings of integers are isomorphic: $\mathbf F_p[x]_Q \cong k[[t]]$. The isomorphism identifies powers of the maximal ideal on both sides, and the maximal ideal of $\mathbf F_p[x]_Q$ is $(Q)$ since $Q$ is a uniformizer in the $Q$-adic completion. Thus $$ \mathbf F_p[x]_Q/(Q^m) \cong k[[t]]/(t^m) $$ for each $m \geq 1$. Both sides simplify to quotient rings of $\mathbf F_p[x]$ and $k[t]$, just as $\mathbf Z_p/(p^m) \cong \mathbf Z/(p^m)$: $$ \mathbf F_p[x]/(Q^m) \cong k[t]/(t^m). $$ Taking $m = e$, we get $$ \mathbf F_p[x]/(Q^e) \cong k[t]/(t^e), $$ so $\mathcal O_K/\mathfrak p^e \cong A/(p) \cong \mathbf F_p[x]/(Q^e) \cong k[t]/(t^e)$.

I remember working this out 25 years ago. The main idea is to view both rings as quotient rings of completions: $\mathcal O_K/\mathfrak p^e \cong \widehat{\mathcal O_{\mathfrak p}}/\widehat{\mathfrak p}^e$ and $k[t]/(t^e) \cong k[[t]]/(t^e)$. Then the structure of the ring of integers of local fields, in characteristic $0$ and characteristic $p$, will help us solve the problem.

The last step is to show $\mathbf F_p[x]/(Q(x)^e) \cong k[t]/(t^e)$ as rings, so $$ \mathcal O_K/\mathfrak p^e \cong A/(p) \cong \mathbf F_p[x]/(Q^e) \cong k[t]/(t^e). $$

In fact we'll show $$ \mathbf F_p[x]/(Q(x)^m) \cong k[t]/(t^m) $$ for all $m \geq 1$. We need $m = e$ only to identify these rings with $\mathcal O_K/\mathfrak p^e$, but the rings are isomorphic to each other for all $m \geq 1$. (Note $\mathcal O_K/\mathfrak p^m$ has characteristic $p$ if and only if $\mathfrak p^m \mid p\mathcal O_K$, forcing $m \leq e$, but the rings $\mathbf F_p[x]/(Q(x)^m)$ and $k[t]/(t^m)$ have characteristic $p$ for all $m \geq 1$, so there's nothing unusual about them winding up as isomorphic to each other for all $m$.) I will describe two methods, the first one being more concrete.

Method 1: The elements of $k[t]/(t^m)$ are uniquely expressible as $$ c_0 + c_1t + \cdots + c_{m-1}t^{m-1} \bmod t^m $$ with $c_j \in k$. Inside $\mathbf F_p[x]/(Q^m)$, the elements are uniquely expressible as $$ a_0(x) + a_1(x)Q(x) + \cdots + a_{m-1}(x)Q(x)^{m-1} $$ with $a_j(x) = 0$ or $\deg(a_j(x)) < \deg Q$. But this way of writing the elements of $\mathbf F_p[x]/(Q^m)$ is terrible in order to set up a ring isomorphism with $k[t]/(t^m)$ since those base $Q$ digits $a_j(x)$ are not multiplicatively closed (in contrast to the $c_j$ in $k$). What we need to do is find a copy of the field $k$ of order $p^f$ inside $\mathbf F_p[x]/(Q^m)$.

The polynomial $t^{p^f} - t$ splits completely over the field $\mathbf F_p[x]/(Q)$, so for each $m \geq 1$ the $Q$-adic Hensel's lemma tells us we can lift each of those roots mod $Q$ uniquely to a root of $t^{p^f}-t$ in $\mathbf F_p[x]/(Q^m)$. Set $$ k_m := \{b(x) \bmod Q^m : b(x)^{p^f} \equiv b(x) \bmod Q^m\}, $$ which is the roots of $t^{p^f}-t$ in $\mathbf F_p[x]/(Q^m)$. This set is closed under addition and multiplication and each nonzero element is invertible: if $b(x) \not\equiv 0 \bmod Q^m$ then $b(x) \not\equiv 0 \bmod Q$ (the only lift of $0 \bmod Q$ as a root is $0 \bmod Q^m$), so $\gcd(b(x),Q^m) = 1$. So $k_m$ is a field of order $p^f$ inside $\mathbf F_p[x]/(Q^m)$ and in fact it's the only such field inside $\mathbf F_p[x]/(Q^m)$ thanks to the unique lifting of roots of $t^{p^f}-t$ from modulus $Q$ to modulus $Q^m$.

Since $k_m$ inside $\mathbf F_p[x]/(Q^m)$ is a set of representatives for $\mathbf F_p[x]/(Q)$, we can write every element of $\mathbf F_p[x]/(Q^m)$ uniquely as $$ b_0(x) + b_1(x)Q(x) + \cdots + b_{m-1}(x)Q(x)^{m-1} \mod Q(x)^m $$ where $b_j(x) \bmod Q(x)^m \in k_m$. This way of writing the elements of $\mathbf F_p[x]/(Q^m)$ looks just like the usual way of writing the elements of $k[t]/(t^m)$ and it shows $k_m[Q \bmod Q^m]$ fills up $\mathbf F_p[x]/(Q^m)$

Since $k$ and $k_m$ are fields of equal size $p^f$ there is a field isomorphism $\varphi_m \colon k \to k_m$ (in fact there are $m$ isomorphisms, but that doesn't matter). Extend $\varphi_m$ to a ring homomorphism $k[t] \to \mathbf F_p[x]/(Q^m)$ by mapping $t$ to $Q \bmod Q^m$: $$ \sum_{i} c_it^i \mapsto \sum_i \varphi_m(c_i)Q^i \bmod Q^m. $$ This is surjective since $\mathbf F_p[x]/(Q^m)$ is generated as a ring by $k_m$ and $Q \bmod Q^m$. Since $t^m \mapsto Q^m \bmod Q^m = 0$, we get an induced surjective ring homomorphism $k[t]/(t^m) \to \mathbf F_p[x]/(Q^m)$. Both have order $p^{fm}$, so this is a ring isomorphism.

Method 2: Since $Q$ is irreducible in $\mathbf F_p[x]$, the ring $\mathbf F_p[x]/(Q^m)$ is unchanged up to isomorphism if we replace $\mathbf F_p[x]$ with its $Q$-adic completion $\mathbf F_p[x]_Q$: $$ \mathbf F_p[x]/(Q^m) \cong \mathbf F_p[x]_Q/(Q^m) $$ as rings. The residue field of $\mathbf F_p[x]_Q$ is isomorphic to $\mathbf F_p[x]/(Q)$, which is isomorphic to $k$. The completion $\mathbf F_p[x]_Q$ is the ring of integers of the $Q$-adic field completion $\mathbf F_p(x)_Q$, and the structure of local fields of positive characteristic says they are all isomorphic to the formal Laurent series field over the residue field. Thus $\mathbf F_p(x)_Q \cong k((t))$ as valued fields, so their rings of integers are isomorphic: $\mathbf F_p[x]_Q \cong k[[t]]$. The isomorphism identifies powers of the maximal ideal on both sides, and the maximal ideal of $\mathbf F_p[x]_Q$ is $(Q)$ since $Q$ is a uniformizer in the $Q$-adic completion. Thus $$ \mathbf F_p[x]_Q/(Q^m) \cong k[[t]]/(t^m) $$ for each $m \geq 1$. Both sides simplify to quotient rings of $\mathbf F_p[x]$ and $k[t]$, just as $\mathbf Z_p/(p^m) \cong \mathbf Z/(p^m)$: $$ \mathbf F_p[x]/(Q^m) \cong k[t]/(t^m). $$

For a nonzero prime ideal $\mathfrak p$ in $\mathcal O_K$ lying over $p$ and $m \geq 1$, the quotient ring $\mathcal O_K/\mathfrak p^m$ is unchanged up to isomorphism if we replace $\mathcal O_K$ by its localization at $\mathfrak p$ or by its completion at $\mathfrak p$ (and the modulus also changes to the ideal it generates in the localization or completion). The ramification index $e = e(\mathfrak p|p)$ and residue field degree $f = f(\mathfrak p|p)$ are unchanged by localizing or completing at $\mathfrak p$.

Let $A$ be the completion of $\mathcal O_K$ at $\mathfrak p$, so in $A$ we can write $p = \pi^e u$ for some uniformizer $\pi$ and $u \in A^\times$. Then $\mathcal O_K/\mathfrak p^e \cong A/(\pi^e) = A/(p)$. (In this step it's important that the exponent $e$ is the ramification index of $\mathfrak p$ over $p$.) The ring $A$ is the ring of integers of the completion $K_\mathfrak p$. Even though $\mathcal O_K$ need not be monogenic, completions are monogenic: the ring of integers of every finite extension of $\mathbf Q_p$ has a power basis over $\mathbf Z_p$, so $A = \mathbf Z_p[\alpha]$ for some $\alpha \in A$. That there is a power basis over $\mathbf Z_p$ is the key fact you were missing.

The main idea is to view both rings as quotient rings of completions: $\mathcal O_K/\mathfrak p^e \cong \widehat{\mathcal O_{\mathfrak p}}/\widehat{\mathfrak p}^e$ and $k[t]/(t^e) \cong k[[t]]/(t^e)$. Then the structure of the ring of integers of local fields, in characteristic $0$ and characteristic $p$, will help us solve the problem.

For a nonzero prime ideal $\mathfrak p$ in $\mathcal O_K$ lying over $p$ and $m \geq 1$, the quotient ring $\mathcal O_K/\mathfrak p^m$ is unchanged up to isomorphism if we replace $\mathcal O_K$ by its localization at $\mathfrak p$ or by its completion at $\mathfrak p$ (and the modulus also changes to the ideal it generates in the localization or completion). Also, the ramification index $e = e(\mathfrak p|p)$ and residue field degree $f = f(\mathfrak p|p)$ are unchanged by localizing or completing at $\mathfrak p$.

Let $A = \widehat{\mathcal O_{\mathfrak p}}$ be the completion of $\mathcal O_K$ at $\mathfrak p$, so in $A$ we can write $p = \pi^e u$ for some uniformizer $\pi$ and $u \in A^\times$. Then $\mathcal O_K/\mathfrak p^e \cong A/(\pi^e) = A/(p)$. (In this step it's important that the exponent $e$ is the ramification index of $\mathfrak p$ over $p$.) The ring $A$ is the ring of integers of the completion $K_\mathfrak p$. Even though $\mathcal O_K$ need not be monogenic, completions are monogenic: the ring of integers of every finite extension of $\mathbf Q_p$ has a power basis over $\mathbf Z_p$, so $A = \mathbf Z_p[\alpha]$ for some $\alpha \in A$. That there is a power basis over $\mathbf Z_p$ is the key fact you were missing.