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.
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