By adapting Kanovei and Shelah's construction of a definable hyperreal field, (EDIT: After actually looking at their paper I feel that I should mention that they basically pointed out this application of their construction at the end of their paper.) I believe I can show that every structure $\mathfrak{A}$ has a definable proper class monster model elementary extension $\mathfrak{C}$ in ZFC without any assumption of global choice (and furthermore the definition is uniform in $\mathfrak{A}$ and does not require a choice of a well-ordering of $\mathfrak{A}$), but please tell me if there's some mistake in my reasoning. It will be proper class saturated in the sense that given any set sized structures $\mathfrak{B}_0 \prec \mathfrak{B}_1\equiv \mathfrak{A}$ and an elementary embedding $f:\mathfrak{B}_0\prec\mathfrak{C}$ there is an elemenetary embedding $g:\mathfrak{B}_1 \prec \mathfrak{C}$ extending $f$. I think you can define a truth predicate on $\mathfrak{C}$ given the elementary diagram of $\mathfrak{A}$, but you don't need it to construct $\mathfrak{C}$.
(I'm basing this off of Keisler's exposition (section 1G) of Kanovei and Shelah's result, rather than the original paper.)
Lemma: There is a uniformly definable family of linear orders $(A_\kappa,\sqsubset_\kappa)$ for each infinite cardinal $\kappa$ and uniformly definable functions $f_\kappa : A_\kappa \rightarrow 2^{2^\kappa}$ such that $f_\kappa(a)$ is a non-principal ultrafilter on $\kappa$ for each $a\in A_\kappa$ and every non-principal ultrafilter on $\kappa$ is $f_\kappa(a)$ for some $a\in A_\kappa$.
Proof: For each infinite cardinal $\kappa$ let $<_\kappa$ be the lexicographical ordering on $2^\kappa$ (i.e. $a <_\kappa b$ if $a(\alpha) < b(\alpha)$ for the first $\alpha$ at which they disagree where $<$ is the standard order on $2=\{0,1\}$). Then let $\sqsubset_\kappa$ be the lexicographic ordering on ${\left(2^\kappa\right)}^\left|2^\kappa\right|$ relative to $<_\kappa$, i.e. the set of functions from $\left|2^\kappa\right|$, the initial ordinal with the same cardinality as $2^\kappa$, to $2^\kappa$ (this is uniform because we don't need to choose a bijection between $2^\kappa$ and $\left|2^\kappa\right|$). Finally let $A_\kappa$ be the set of all functions $g\in {\left(2^\kappa\right)}^\left|2^\kappa\right|$ whose range is a non-principal ultrafilter on index set $\kappa$. Then $f_\kappa(g)=\text{range}(g)$. $\square$
So now we're going to define a proper class length elementary chain of elementary extensions of $\mathfrak{A}$. Let $\mathfrak{C}_0 = \mathfrak{A}$. For limit ordinals $\lambda$, let $\mathfrak{C}_\lambda = \bigcup_{\alpha<\lambda} \mathfrak{C}_\alpha$. Otherwise let $\mathfrak{C}^\prime_\alpha$ be the finite support iterated ultrapower of $\mathfrak{C}_\alpha$ using $A_{\aleph_{\alpha}}$. If $h:\mathfrak{C}_\alpha \rightarrow \mathfrak{C}^\prime_\alpha$ is the natural embedding, let $\mathfrak{C}_{\alpha+1}$ be $\mathfrak{C}_\alpha \cup \left( \mathfrak{C}_\alpha^\prime \backslash h(\mathfrak{C}_\alpha) \right)$ and define all the atomic predicates accordingly.
So then let $\mathfrak{C}= \bigcup_{\alpha\in\mathbf{Ord}}\mathfrak{C}_\alpha$. If $\mathfrak{B}_0 \prec \mathfrak{B}_1$ are set sized and $\mathfrak{B}_0 \prec \mathfrak{C}_\alpha$ for some $\alpha$, then by the typical kind of argument there exists an ultrafilter $U$ on some cardinal $\aleph_\beta$ such that $\mathfrak{B}_1$ embeds into $\prod_{U}\mathfrak{C}_\alpha$ in a way that fixes $\mathfrak{B}_0$. By padding the index set we may assume that $\beta \geq \alpha$, so we have that at some point in the chain, $\mathfrak{C}_\beta$ contains an elementary substructure isomorphic to $\prod_{U}\mathfrak{C}_\alpha$ in a way that fixes $\mathfrak{C}_\alpha$, so we get that $\mathfrak{B}_1$ can be embedded into $\mathfrak{C}_\beta$ in a way that fixes $\mathfrak{B}_0$.
More or less $\mathfrak{C}$ is the finite support iterated ultrapower of $\mathfrak{A}$ along the linear order $\sum_{\kappa\in\textbf{InfCard}}(A_\kappa,\sqsubset_\kappa)$, similar to what Joel suggested.
We also get a certain amount of homogeneity in that any automorphism of some $\mathfrak{C}_\alpha$ extends to an automorphism of all of $\mathfrak{C}$ (in a way that's uniformly definable in terms of the automorphism, because really any expansion of the theory at some $\mathfrak{C}_\alpha$ uniformly extends to all of $\mathfrak{C}$). I think that at certain strong limit cardinals (i.e. $\aleph_\alpha = \beth_\lambda$ for limit ordinal $\lambda$), $\mathfrak{C}_{\alpha}$ will be a special model of the theory, and therefore resplendent, so given any partial elementary map $h:A\rightarrow B$ for sets $A,B\subset \mathfrak{C}$, I think it can be extended to an automorphism of some $\mathfrak{C}_{\alpha}$ for sufficiently large $\aleph_\alpha = \beth_\lambda$ (in a non-uniform way that required choosing an ordering of $\mathfrak{C}_{\alpha}$), which can then be extended to a definable (with parameters) automorphism of $\mathfrak{C}$.
