We present a fast scheme for arbitrary unitary control of interacting bosonic atoms in a double well. Assuming fixed interwell tunneling rate and intrawell interaction strength, we control the many-atom state by a discrete sequence of shifts of the single-well energies. For strong interactions, resonant tunneling transitions implement beam-splitter U(2) rotations among atom number eigenstates, which can be combined and, thus, permit full controllability. By numerically optimizing such sequences of couplings at avoided level crossings, we extend the realm of full controllability to a wide range of realistic interaction parameters, while we remain in the simple control space. We demonstrate the efficiency and the high achievable fidelity of our proposal with nonadiabatic population transfer, NOON-state creation, a cnot gate, and a transistorlike, conditional evolution of several atoms.