I have a result for graphs whose stability number=clique cover number, which naturally includes the perfect graphs, but I'm curious about if there are other known and well-definable graph classes which fit the bill.
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Let $G$ be a $k$-regular graph with a 1-factorization. Then its line graph $L(G)$ is $k$-colorable and has cliques of size $k$, and so its complement will be in your class if it's not perfect. It will not be perfect if the girth of $G$ is five, because an induced 5-cycle in $G$ gives an induced 5-cycle in $L(G)$, and in its complement.