We report on two millimeter flares detected by ALMA at 220 GHz from AU Mic, a nearby M dwarf. The larger flare had a duration of only $sim35$ sec, with peak $L_{R}=2times10^{15}$ erg s$^{-1}$ Hz$^{-1}$, and lower limit on linear polarization of $|Q/I|>0.12pm0.04$. We examine the characteristics common to these new AU Mic events and those from Proxima Cen previously reported in MacGregor et al. (2018) - namely short durations, negative spectral indices, and significant linear polarization - to provide new diagnostics of conditions in outer stellar atmospheres and details of stellar flare particle acceleration. The event rates ($sim20$ and $4$ events day$^{-1}$ for AU Mic and Proxima Cen, respectively) suggest that millimeter flares occur commonly but have been undetected until now. Analysis of the flare observing frequency and consideration of possible incoherent emission mechanisms confirms the presence of MeV electrons in the stellar atmosphere occurring as part of the flare process. The spectral indices point to a hard distribution of electrons. The short durations and lack of pronounced exponential decay in the light curve are consistent with formation in a simple magnetic loop, with radio emission predominating from directly precipitating electrons. We consider the possibility of both synchrotron and gyrosynchrotron emission mechanisms, although synchrotron is favored given the linear polarization signal. This would imply that the emission must be occurring in a low density environment of only modest magnetic field strength. A deeper understanding of this newly discovered and apparently common stellar flare mechanism awaits more observations with better-studied flare components at other wavelengths.
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