As the simplest two-dimensional (2D) polymer, graphene has immensely high intrinsic strength and elastic stiffness but has limited toughness due to brittle fracture. We use atomistic simulations to explore a new class of graphene/polyethylene hybrid 2D polymer, "graphylene", that exhibits ductile fracture mechanisms and has a higher fracture toughness and flaw tolerance than graphene. A specific configuration of this 2D polymer hybrid, denoted "GrE-2" for the two-carbon-long ethylene chains connecting benzene rings in the inherent framework, is prioritized for study. MD simulations of crack propagation show that the energy release rate to propagate a crack in GrE-2 is twice that of graphene. We also demonstrate that GrE-2 exhibits delocalized failure and other energy-dissipating fracture mechanisms such as crack branching and bridging. These results demonstrate that 2D polymers can be uniquely tailored to achieve a balance of fracture toughness with mechanical stiffness and strength.