If we consider human manipulation, it is clear that contact-rich manipulation (CRM)-the ability to use any surface of the manipulator to make contact with objects-can be far more efficient and natural than relying solely on end-effectors (i.e., fingertips). However, state-of-the-art model-based planners for CRM are still focused on feasibility rather than optimality, limiting their ability to fully exploit CRM's advantages. We introduce a new paradigm that computes approximately optimal manipulator plans. This approach has two phases. Offline, we construct a graph of mutual reachable sets, where each set contains all object orientations reachable from a starting object orientation and grasp. Online, we plan over this graph, effectively computing and sequencing local plans for globally optimized motion. On a challenging, representative contact-rich task, our approach outperforms a leading planner, reducing task cost by 61%. It also achieves a 91% success rate across 250 queries and maintains sub-minute query times, ultimately demonstrating that globally optimized contact-rich manipulation is now practical for real-world tasks.