We present numerical solutions of the problem of non-relativistic, radiation mediated shock (RMS) breakouts from optically thick circumstellar winds. Such breakouts are expected to occur in a significant fraction of supernova explosions. Our solutions improve on earlier work in several major respects: (i) They include a description of the changes in photons’ energy due to inelastic Compton scattering, which plays a crucial role in determining the radiation spectrum; (ii) They include a description of the collisionless shock formed following breakout; (iii) They include a self-consistent calculation of the interaction of the radiation escaping the shock with the upstream un-shocked plasma. A novel numerical scheme is developed to solve the coupled multi-group radiation-hydrodynamics equations and is shown to be both stable and convergent. We find that for a typical shock breakout velocity of v/c=0.03  the spectrum develops a hard X-ray component carrying a significant fraction of the luminosity, peaking at ~3 keV shortly after the peak breakout bolometric luminosity and shifting at later time to higher X-ray energies.