Current observations with the James Webb Space Telescope (JWST) suggest that star-forming galaxies produce enough ionizing (LyC; $\lambda < 91.2$ nm) photons to drive cosmic reionization, but the efficiency with which these photons escape their host galaxies remains uncertain. Absorption by the neutral intergalactic medium progressively suppresses direct LyC detections above redshift $z\sim3$, forcing astronomers to rely on indirect diagnostics of LyC escape calibrated at low redshift. Low-resolution ultraviolet observations of high-redshift analogs obtained with the Cosmic Origins Spectrograph onboard the Hubble Space Telescope (HST) have been critical for developing these diagnostics. These studies suggest that stellar feedback plays a central role in regulating LyC escape, although the role of galactic winds and the underlying physical mechanisms remain poorly constrained. High-resolution spectroscopy blueward of 160.0 nm (rest-frame) is required to resolve the kinematic structure of the winds and reveal the physics governing LyC escape. Such observations are currently only possible with HST and represent a major science driver for the future Habitable Worlds Observatory (HWO). Extending the lifetime of HST and prioritizing ultraviolet observations are essential for interpreting current JWST studies of the early Universe and important preparatory science for HWO.