  Recent space-based missions have ushered in a new era of observational
astronomy, where high-cadence photometric light curves for thousands to
millions of stars in the solar neighborhood can be used to test and apply
stellar age-dating methods, including gyrochronology. Combined with precise
kinematics, these data allow for powerful new insights into our understanding
of the Milky Way's dynamical history. Using TESS data, we build a series of
rotation period measurement and confirmation pipelines and test them on 1,560
stars across five benchmark samples: the Pleiades, Pisces--Eridanus, Praesepe,
the Hyades, and field stars from the MEarth Project. Our pipelines' recovery
rates across these groups are on average 89\%. We then apply these pipelines to
4,085 likely single stars with TESS light curves in two interesting regions of
Galactic action space. We identify 141 unique, rapidly rotating stars in highly
eccentric orbits in the disk, some of which appear as rotationally young as the
120-Myr-old Pleiades. Pending spectroscopic analysis to confirm their youth,
this indicates these stars were subject to fast-acting dynamical phenomena, the
origin of which will be investigated in later papers in this series.
