Is there a definable model of PA whose domain is a proper class and whose complete theory is not definable?

Assume ZFC. Is there a formula of $mathcal{L}_in$ (without parameters) defining a model $mathcal{M}$ of PA whose domain is a proper class but the complete theory of that model is not definable by any formula of $mathcal{L}_in$ (again without parameters)?

Edit: Instead of model, I should really have said relative interpretation. So I want five formulas $psi_D(cdot)$, $psi_0(cdot)$, $psi_1(cdot)$,$psi_+(cdot)$ and $psi_times(cdot)$ to define proper classes M, $0^mathcal{M}$,$1^mathcal M$,$+^mathcal M$ and $times^mathcal M$. With some abuse of notation I’ll write

$$mathcal M = langle M, 0^mathcal M ,1^mathcal M ,+^mathcal M,times ^mathcal M rangle. $$

Then when I say that $mathcal M $ is a model so PA, I mean that $varphi^mathcal M$ for every axiom $varphi$ of PA.