This is extremely contrived, but it seems quite close to what you're looking for.
Suppose $M$ is transitive model of ZFC and $P\in M$. We want to describe a process for adding a subset $G$ of $P$ to $M$, and in the end we want this process to yield a model $N_G$ with $M\cup \{G\}\subseteq N_G$ and $\text{Ord}\cap M = \text{Ord}\cap N_G$.
Obviously not just any set $G$ can be adjoined to $M$ while preserving ZFC, so we our process will only apply to certain "good" sets $G$. We have to figure out what these good sets are.
If one can somehow approximately describe the extension $N_G$ within $M$, one might hope it is possible to propagate ZFC from $M$ to $N_G$. So we want our good sets to be the ones that permit such a description. What would it mean to describe $N_G$?
Well, to start, we should have a definable subclass $M^P$ of $M$ that we can view as terms for the elements of the model $N_G$ we are trying to describe (although of course, we cannot hope to have complete information about membership relation on the terms $M^P$ within $M$). Each good $G$ should yield a way to evaluate these terms and obtain an element of $N_G$. Let $i_G : M^P\to N_G$ be this interpretation function. Of course, $i_G$ should be a surjection.
The structure of $N_G$ should only depend on which elements of $P$ belong to $G$, so we might hope that for any terms $\vec \tau\subseteq M^P$, any elements $\vec a\subseteq M$, any formula $\varphi$, and any good $G$, if $N_G\vDash \varphi(i_G(\vec \tau),\vec a)$ then in fact there is some $p\in G$ such that for any good $H$ with $p\in H$, $N_H \vDash \varphi(i_H(\vec \tau),\vec a)$. Call this the truth hypothesis.
Also if $M$ is to approximate the structure of $N_G$, we might hope that $M$ can see which facts occur no matter which good $G$ we are trying to adjoin. So we demand that for any terms $\vec \tau\subseteq M^P$, any elements $\vec a\subseteq M$ and any formula $\psi$, the class $$\{(\sigma,b)\in M^P\times M : \text{for all good $G$, }N_G\vDash \psi(i_G(\vec \tau),\vec a,G,\sigma,b)\}$$ is definable over $M$. Call this the definability hypothesis.
These hypotheses alone imply that for every good $G$, the model $N_G$ is a set-generic extension of $M$. To see this, it suffices by a recently famous theorem of Bukovsky to show that $(M,N_G)$ has the uniform covering property: for any ordinal $\alpha\in M$ and any function $f : \alpha\to \text{Ord}$ in $N_G$, there is some $F : \alpha\to P(\beta)$ in $M$ such that for all $\xi < \alpha$, $|F(\xi)| \leq |P|$ in $M$ and $f(\xi)\in F(\xi)$.
Fix $f : \alpha\to \text{Ord}$ in $N_G$, and we will define a uniform cover in $M$. Take a term $\tau\in M^P$ with $i_G(\tau) = f$. For each $\xi < \alpha$, let $$F(\xi) = \{\beta : \text{for some $p\in P$, for all good $H$ with $p\in H$}, i_H(\tau)(\xi) = \beta\}$$
Fix $\xi < \alpha$. We know $F(\xi)$ is a definable class of $M$ by the definability hypothesis. Moreover, the definability hypothesis implies the $M$-definability of the partial function $f_\xi : P\to F(\xi)$ given by setting $f_\xi(p) = \beta$ if for all good $H$ with $p\in H$, $i_H(\tau)(\xi) = \beta$. By the truth hypothesis, for each $\beta\in F(\xi)$, there is some $p\in G$ such that for all good $H$ with $p\in H$, $i_H(\tau)(\xi) = \beta$, or in other words, $f_\xi(p) = \beta$. Thus $f_\xi$ is a partial surjection from $P$ to $F(\xi)$, so $|F(\xi)|\leq |P|$ in $M$. This establishes the Bukovsky criterion.
Now replace the assumption that $N_G$ satisfies ZFC with the assumption that for all $S\in M$, $\{i_G(\sigma) : p\in G\text{ and }(\sigma,p)\in S\}$ belongs to $N_G$. (Clearly this must be true if $N_G$ is to satisfy ZFC, assuming $i_G$ is definable over $N_G$.) Then I think you can prove that $N_G$ is a model of ZFC. For example, suppose that $A\in N_G$ and $\varphi$ is a formula and $\vec c\subseteq N_G$ are some parameters. We'll prove the instance of separation for $A$ and $\varphi$. Take terms $\dot A$ for $A$ and $\vec \tau$ for $c$. By the definability hypothesis, the set $S = \{(\sigma,p) : \text{for all good $H$ with $p\in H$}, N_G\vDash \sigma\in \tau\text{ and }\varphi(\sigma)\}$ belongs to $M$. By the truth hypothesis, $\{x\in A : N_G\vDash \varphi(x)\} = \{i_G(\sigma) : p\in G\text{ and }(\sigma,p)\in S\}$, which we assumed was in $N_G$.