When $\mathfrak{g}$ is non-zero and finite dimensional over $k$, its restricted enveloping algebra $u(\mathfrak{g})$ is never a domain. To see this, note that $u(\mathfrak{g})$ is itself finite dimensional over $k$ (of dimension $p^{\dim \mathfrak{g}}$ by Jacobson's PBW theorem) so if it's a domain it must be a division ring. But the definition of $u(\mathfrak{g})$ by generators and relations shows that $u(\mathfrak{g})$ always has a proper ideal --- the augmentation ideal $\mathfrak{g} u(\mathfrak{g})$. This is a contradiction.
When $\mathfrak{g}$ is infinite dimensional over $k$, it can happen that $u(\mathfrak{g})$ is a finitely generated domain. Here is an example: let $\mathfrak{g}$ be the $k$-linear span of the monomials $X, X^p, X^{p^2}, X^{p^3}, \ldots$ inside the polynomial algebra $k[X]$ and view it as an abelian restricted Lie algebra with the obvious restricted structure $x^{[p]} = x^p$ for all $x \in \mathfrak{g}$ (the latter calculated inside $k[X]$). It turns out that actually $u(\mathfrak{g})$ is isomorphic to $k[X]$ in this case, hence $u(\mathfrak{g})$ is a domain.
Note that if $p = 2$ and we view $k[X]$ as a graded ring with $\deg X = 2$, then $\mathfrak{g}$ is a graded restricted Lie algebra concentrated in even positive degrees and finite dimensional in each degree --- i.e. it satisfies your conditions.
Starting from an arbitrary group $G$, it is possible to cook up a restricted graded Lie algebra $\mathfrak{g} = \bigoplus_{n=1}^\infty (D_n/D_{n+1}) \otimes_{\mathbb{F}_p}k$. Here $D_1 \geq D_2 \geq D_3 \geq \cdots $ is the so-called modular dimension series of $G$, defined by Lazard's closed formula
$D_n = \prod_{ip^j \geq n} \gamma_i(G)^{p^j}$
where $\gamma_i(G)$ is the lower central series of $G$ defined by $\gamma_1(G) = G$ and $\gamma_i(G) = [G, \gamma_{i-1}(G)]$ for all $i \geq 2$. The $p$-power operation on $\mathfrak{g}$ is induced by the $p$-power map on the group. Then the theorem of S.A.Jennings asserts that $u(\mathfrak{g})$ is isomorphic to the associated graded ring of the group algebra $k[G]$ with respect to the filtration by powers of the augmentation ideal $I$ of $k[G]$:
$u(\mathfrak{g}) \cong \bigoplus_{n=0}^\infty \frac{I^n}{I^{n+1}}$.
The example I gave above is obtained by taking $G = \mathbb{Z}$. Similar examples can be obtained by taking $G$ to be a uniform pro-$p$ group of rank $d$; then $\mathfrak{g}$ will be an abelian, restricted graded Lie algebra with $u(\mathfrak{g})$ isomorphic to the polynomial algebra $k[X_1,\ldots, X_d]$.
You can find a good account of this material in Chapters 11 and 12 of the book "Analytic pro-$p$ groups" by Dixon, Du Sautoy, Mann and Segal.
