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We propose to use long-slit STIS-G140M spectroscopy to spatially resolve red giant winds for the first time. Specifically comma we will study the H I Lyman-alpha wind scattering around two red giant stars comma Alpha Tau (K5 III) and Gamma Cru (M3.5 III). This project is stimulated by the very recent realization that in existing STIS-E140M spectra of these giants and others comma the Lyman-alpha emission is to a large extent uniformly filling the 0.2x0.2 arcsecond STIS aperture comma especially for the M giants. The only obvious way to explain this surprising result is to propose that H is actually ionized close to the stars comma only becoming neutral far from the star (>10 R_star). Such a picture could dramatically alter our understanding of red giant winds comma potentially increasing mass loss rate estimates for such stars by over an order of magnitude. Our proposed new observations will determine exactly how far the Lyman-alpha scattering region truly extends around K and M giants. In combination with constraints on the winds from existing STIS echelle spectra comma we will be able to make more accurate mass loss rate measurements comma involving better constrained assessments of wind temperature and ionization structure. Finally comma as these winds are known to be variable comma we could potentially be able to detect and study spatial structure in red giant winds for the first time.