Question: Is there a model $M$ of ${\sf ZFC}$ with two satisfaction classes $S$ and $S'$ such that for some formula $\phi$ and assignment $a$ in $M$, $\{x: \langle\phi, a[0/x]\rangle\in S\} = S'$ (where $a[0/x]$ is the assignment according to $M$ which is just like $a$ except that it assigns $x$ to the variable $x_0$)?

Obviously, $M$ will have to be non-standard, as will $\phi$.

I'm primarily interested in models of ${\sf ZFC}$, but I'd also be interested in models of ${\sf PA}$.