We present results from numerical evolutions of single black holes distorted by axisymmetric, but equatorially asymmetric, gravitational (Brill) waves. Net radiated energies, apparent horizon embeddings, and recoil velocities are shown for a range of Brill wave parameters, including both even and odd parity distortions of Schwarzschild black holes. We find that a wave packet initially concentrated on the black hole throat, a likely model also for highly asymmetric stellar collapse and late stage binary mergers, can generate a maximum recoil velocity of about 150 (23) km/sec for even (odd) parity perturbations, significantly less than that required to eject black holes from galactic cores. [S0556-2821(99)10418-1].