The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) is consistent.

Wojowu's answer already points out that the proof of the following fact (S) can be found in the paper Pointwise Definable Models of Set Theory (JSL 2013) by Hamkins, Linetsky, and Reitz.

(S) Every countable model of ZFC has a (class) generic extension to model of ZFC that is pointwise definable.

[Historical note: (S) was asserted with a proof outline in my paper Models of set theory with definable ordinals, which includes a detailed proof of the weaker statement: every countable model of ZFC has a generic extension in which all the ordinals are parameter-free definable. The paper of Hamkins-Linetsky-Reitz presents a detailed proof of not only (S), but also of a corresponding result for countable models of GBC].

Note that (S) is a theorem of ZFC (indeed, it is provable in much weaker systems , but that is a different story). It is easy to see that if the theory ZFC + Con(ZFC) is consistent, then so is $T$ = ZFC + Con(ZFC) + V = L. The fact that $T$ includes the statement V = L implies that $T$ has a global well-ordering of its ambient universe and therefore $T$ has definable Skolem functions. This immediately implies that $T$ has a pointwise definable model $M$. Since (S) is a theorem of ZFC, it holds in $M$, and moreover EVERYTHING in $M$ is parameter-free definable. Thus $M$ is a model in which every countable $N$ model of ZFC has a (class) generic extension $N[G]$ such that (1) $N(G)$ is pointwise definable, and (2) $N(G)$ is parameter-free definable in $M$.

The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) is consistent.

Wojowu's answer already points out that the proof of the following fact (S) can be found in the paper Pointwise Definable Models of Set Theory (JSL 2013) by Hamkins, Linetsky, and Reitz.

(S) Every countable model of ZFC has a (class) generic extension to model of ZFC that is pointwise definable.

[Historical note: (S) was asserted with a proof outline in my paper Models of set theory with definable ordinals, which includes a detailed proof of the weaker statement: every countable model of ZFC has a generic extension in which all the ordinals are parameter-free definable. The paper of Hamkins-Linetsky-Reitz presents a detailed proof of not only (S), but also of a corresponding result for countable models of GBC].

Note that (S) is a theorem of ZFC (indeed, it is provable in much weaker systems , but that is a different story). It is easy to see that if the theory ZFC + Con(ZFC) is consistent, then so is $T$ = ZFC + Con(ZFC) + V = L. The fact that $T$ includes the statement V = L implies that $T$ has a global well-ordering of its ambient universe and therefore $T$ has definable Skolem functions. This immediately implies that $T$ has a pointwise definable model $M$. Since (S) is a theorem of ZFC, it holds in $M$, and moreover EVERYTHING in $M$ is parameter-free definable. Thus $M$ is a model in which every countable $N$ model of ZFC has a (class) generic extension $N[G]$ such that (1) $N[G]$ is pointwise definable, and (2) $N[G]$ is parameter-free definable in $M$.

The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) is consistent.

Wojowu's answer already points out that the a proof of the following fact (S) can be found in the paper Pointwise Definable Models of Set Theory (JSL 2013) by Hamkins, Linetsky, and Reitz.

(S) Every countable model of ZFC has a (class) generic extension to model of ZFC that is pointwise definable.

[Historical note: (S) was asserted with a proof outline in my paper Models of set theory with definable ordinals, which includes a detailed proof of the weaker statement: every countable model of ZFC has a generic extension in which all the ordinals are parameter-free definable. The paper of Hamkins-Linetsky-Reitz presents a detailed proof of not only (S), but a corresponding result for countable models of GBC].

Note that (S) is a theorem of ZFC (indeed, it is provable in much weaker systems , but that is a different story). It is easy to see that if the theory ZFC + Con(ZFC) is consistent, then so is $T$ = ZFC + Con(ZFC) + V = L. The fact that $T$ includes the statement V = L implies that $T$ has a global well-ordering of its ambient universe and therefore $T$ has definable Skolem functions. This immediately implies that $T$ has a pointwise definable model $M$. Since (S) is a theorem of ZFC, it holds in $M$, and moreover EVERYTHING in $M$ is parameter-free definable. Thus $M$ is a model in which every countable $N$ model of ZFC has a (class) generic extension $N[G]$ such that (1) $N(G)$ is pointwise definable, and (2) $N(G)$ is parameter-free definable in $M$.

The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) is consistent.

Wojowu's answer already points out that the proof of the following fact (S) can be found in the paper Pointwise Definable Models of Set Theory (JSL 2013) by Hamkins, Linetsky, and Reitz.

(S) Every countable model of ZFC has a (class) generic extension to model of ZFC that is pointwise definable.

[Historical note: (S) was asserted with a proof outline in my paper Models of set theory with definable ordinals, which includes a detailed proof of the weaker statement: every countable model of ZFC has a generic extension in which all the ordinals are parameter-free definable. The paper of Hamkins-Linetsky-Reitz presents a detailed proof of not only (S), but also of a corresponding result for countable models of GBC].

Note that (S) is a theorem of ZFC (indeed, it is provable in much weaker systems , but that is a different story). It is easy to see that if the theory ZFC + Con(ZFC) is consistent, then so is $T$ = ZFC + Con(ZFC) + V = L. The fact that $T$ includes the statement V = L implies that $T$ has a global well-ordering of its ambient universe and therefore $T$ has definable Skolem functions. This immediately implies that $T$ has a pointwise definable model $M$. Since (S) is a theorem of ZFC, it holds in $M$, and moreover EVERYTHING in $M$ is parameter-free definable. Thus $M$ is a model in which every countable $N$ model of ZFC has a (class) generic extension $N[G]$ such that (1) $N(G)$ is pointwise definable, and (2) $N(G)$ is parameter-free definable in $M$.