The properties of graphene open new opportunities for the fabrication of composites exhibiting unique structural and functional capabilities. However, to achieve this goal we should build materials with carefully designed architectures. Here, we describe the fabrication of ceramic-graphene composites by combining graphene foams with pre-ceramic polymers and spark plasma sintering. The result is a material containing an interconnected, microscopic network of very thin (20-30 nm), electrically conductive, carbon interfaces. This network generates electrical conductivities up to two orders of magnitude higher than those of other ceramics with similar graphene or carbon nanotube contents and can be used to monitor 'in situ' structural integrity. In addition, it directs crack propagation, promoting stable crack growth and increasing the fracture resistance by an order of magnitude. These results demonstrate that the rational integration of nanomaterials could be a fruitful path towards building composites combining unique mechanical and functional performances.