  We have searched for short periodicities in the light curves of stars with
$T_{\rm eff}$ cooler than 4000 K made from 2-minute cadence data obtained in
TESS sectors 1 and 2. Herein we report the discovery of 10 rapidly rotating
M-dwarfs with highly structured rotational modulation patterns among 10 M
dwarfs found to have rotation periods less than 1 day. Star-spot models cannot
explain the highly structured periodic variations which typically exhibit
between 10 and 40 Fourier harmonics. A similar set of objects was previously
reported following K2 observations of the Upper Scorpius association (Stauffer
et al. 2017). We examine the possibility that the unusual structured
light-curves could stem from absorption by charged dust particles that are
trapped in or near the stellar magnetosphere. We also briefly explore the
possibilities that the sharp structured features in the lightcurves are
produced by extinction by coronal gas, by beaming of the radiation emitted from
the stellar surface, or by occultations of spots by a dusty ring that surrounds
the star. The latter is perhaps the most promising of these scenarios. Most of
the structured rotators display flaring activity, and we investigate changes in
the modulation pattern following the largest flares. As part of this study, we
also report the discovery of 371 rapidly rotating M-dwarfs with rotational
periods below 4 hr, of which the shortest period is 1.63 hr.
