  We report the serendipitous detection by GALEX of fast (<1 day) rising (>1
mag) UV emission from two Type II plateau (II-P) supernovae (SNe) at z=0.185
and 0.324 discovered by the Supernova Legacy Survey. Optical photometry and VLT
spectroscopy 2 weeks after the GALEX detections link the onset of UV emission
to the time of shock breakout. Using radiation hydrodynamics and non-LTE
radiative transfer simulations, and starting from a standard red supergiant
(RSG; Type II-P SN progenitor) star evolved self-consistently from the main
sequence to iron core collapse, we model the shock breakout phase and the 55 hr
that follow. The small scale height of our RSG atmosphere model suggests that
the breakout signature is a thermal soft X-ray burst (lambda_peak ~ 90\AA) with
a duration of <~ 2000 s. Longer durations are possible but require either an
extended and tenuous non-standard envelope, or an unusually dense RSG wind with
\dot{M} ~ 10^(-3) Msun yr^(-1). The GALEX observations miss the peak of the
luminous (M_FUV ~ -20) UV burst but unambiguously capture the rise of the
emission and a subsequent 2 day long plateau. The postbreakout, UV-bright
plateau is a prediction of our model in which the shift of the peak of the
spectral energy distribution (SED) from ~100 to ~1000\AA and the ejecta
expansion both counteract the decrease in bolometric luminosity from ~10^11 to
~10^9 L_sun over that period. Based on the observed detection efficiency of our
study we make predictions for the breakout detection rate of the GALEX Time
Domain Survey.
