We present a generic resynthesis framework for optimizing Boolean networks parameterized with a multi-level logic representation, a cut-computation algorithm, and a resynthesis algorithm. The framework allows us to realize powerful optimization algorithms in a plug-and-play fashion. We show the framework's versatility by composing an exact DAG-aware rewriting engine. Disjoint-support decomposition and SAT-based exact synthesis together with efficient caching strategies enable the algorithm to resynthesize larger parts of the logic. DAG-aware rewriting is used to compute the gain of resynthesis while taking the benefit of structural hashing into account.