In this paper, we propose a new global router, BOB-Router, endowed with the ability to use over-the-block routing resources to the greatest extent in addition to traditional routing concepts of minimizing wirelength, via count and overflow. In previous global routing formulations, the routing resources over the IP blocks were either dealt as routing blockages leading to a significant waste, or simply treated in the same way as outside-the-block routing resources, which violates the slew constraints and thus fail buffering. Utilizing over-the-block routing resources could dramatically improve the routing solution, yet requires special attention, since the slew, affected by different RC on different metal layers, must be constrained by buffering and is easily violated. Moreover, even all nets are slew-legalized, the routing solution could still suffer from heavy congestion problem. For the first time, BOB-Router tries to solve the over-the-block global routing problem through minimizing overflows, wirelength and via count simultaneously without violating slew constraints. BOB-Router generates a slew-legalized initial solution followed by a Lagrangian-multiplier-based pricing phase and RC-constrained A* search to help explore new buffering-aware topologies on all metal layers. Our experimental results show that BOB-Router completely satisfies the slew constraints and significantly outperforms the obstacle-avoiding global routers in terms of wirelength, via count and overflows.