Every ordinal $α$ that is the least ordinal satisfying a given $Σ^1_1$ property (about $α$, equivalently, about $L_α$) is a Gandy ordinal, so $\mathsf{Def}(L_α)=\mathsf{Ad}(L_α)$. This includes the least $L_α$ satisfying ZFC, and even includes the least $L_α$ with $α$ an inaccessible cardinal in $L_{α^{+\text{CK}}}$ (but using $L_{α^{+\text{CK}}+1}$ would fail).
The proof of being Gandy is analogous to the proof of existence of recursive pseudowellorderings. Consider a theory $S$ including $\mathrm{ATR}_0$ and "ordinal $α$ satisfies a given $Σ^1_1$ property", with definable Skolem functions. Let $T$ be the tree of (essentially) finite partial $α$-models (as in $ω$-models) of $S$: A sequence of ordinals $<α$ $x_1,x_2,...,x_n$ is in $T$ iff there is no inconsistency proof shorter than $n$ for "$S$, symbols $x_1,x_2,...,x_n$, ordering of $x_1,x_2,...,x_n$, and $∃y<α \, φ(y) ⇒ φ(x_{⌈φ⌉})$ for the first $n$ one-variable $S$-formulas $φ$" ($⌈φ⌉$ is the Gödel number). Then the Kleene–Brouwer order of $T$ has a well-ordered initial segment of length $α^{+\text{CK}}$. The existence of the desired model ensures that the order is ill-founded, while its nonexistence in $L_{α^{+\text{CK}}}$ (since per assumption, $α$ is the least ordinal with the $Σ^1_1$ property, and we use $\mathrm{ATR}_0$) ensures that the well-founded length equals $α^{+\text{CK}}$.
As an aside, this result may seem surprising since working in ZFC or NBG, we can develop a theory of proper class well-orderings, and using replacement, failure of well-foundedness would be witnessed as a set. By diagonalization, increasing descriptive complexity gives strictly longer proper class well-orderings (at least for parameter-free complexity or if $V=HOD$), with 'longer' interpreted using embeddability into proper initial segments (which makes sense even though I think not all first-order definable class well-orderings are comparable using first-order definable classes). And for extensions of ZFC using infinitary logic or constructible hierarchy above $V$, the increase in well-ordering lengths continues transfinitely with even the supremum of $L_\mathrm{Ord}(V)$ well-orderings not reaching the height of $\mathsf{Ad}(V)$ (assuming correctness about well-foundedness). But as soon as replacement (for countable sequences) fails (which presumably it would not for the 'true' $V$), the supposed well-orderings can turn out to be an illusion.