The striking nonlinear effects exhibited by cavity QED systems make them a powerful tool in modern condensed matter and atomic physics. A recently discovered example is the quantized pumping of energy into a cavity by a strongly coupled, periodically driven spin. We uncover a remarkable feature of these energy pumps: they coherently translate, or boost, a quantum state of the cavity in the Fock basis. Current optical cavity and circuit QED experiments can realize the required Hamiltonian in a rotating frame. Boosting thus enables the preparation of highly excited nonclassical cavity states in near-term experiments.