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 $<α$ &nbsp; $x_1,x_2,...,x_n$ 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 formulas $φ$" ($⌈φ⌉$ is the Gödel number; $x_n$ is not used if $⌈φ⌉≤n$).  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 $α^{+\text{CK}}$.<br/>
In fact (not needed here), there is a recursive order-invariant linear order $f$ on $ℚ^{<ω}$ such that for all $α$ that are the least ordinal for some $Σ^1_1$ property, the initial well-founded portion of $f↾α^{<ω}$ has length $α^{+\mathrm{CK}}$ (by order invariance it does not matter how $α$ is order-preservingly embedded into $ℚ$).  The construction is to interleave separate partial models as above for all variations of the theory $S$, except that we can skip variations at the cost of choosing a higher branch.

A related result is that for every finite $n$, $\mathsf{Def}(L_α)=\mathsf{Ad}(L_α)$ for the minimum $α$ such that $L_α$ satisfies ZFC and is correct about well-foundedness of $\mathbf{Σ}_n^{L_α}$ relations.  The proof is as above except that we require that the partial models are correct about $Σ_{n+1}^{L_α}$ properties of the ordinals.  This way a resulting model is correct about well-foundedness of relevant relations, and hence cannot be shorter than $α$.

As an aside, the results 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 (including the replacement schema) 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' $V$), in some cases, the supposed well-orderings may turn out to be an illusion.