We have obtained long-slit spectra of 174 star-forming galaxies with stellar masses greater than 1010 M ☉ from the GALEX Arecibo Sloan Digital Sky Survey (GASS) survey. These galaxies have both H I and H2 mass measurements. The average metallicity profile is strikingly flat out to R 90, the radius enclosing 90% of the r-band light. Metallicity profiles which decline steadily with radius are found primarily for galaxies in our sample with low stellar mass (log(M *) < 10.2), concentration, and/or mean stellar mass density. Beyond ~R 90, however, around 10% of the galaxies in our sample exhibit a sharp downturn in metallicity. Remarkably, we find that the magnitude of the outer metallicity drop is well correlated with the total H I content of the galaxy (measured as f H I = M H I /M *). We examine the radial profiles of stellar population ages and star formation rate densities, and conclude that the galaxies with largest outer metallicity drops are actively growing their stellar disks, with mass-doubling times across the whole disk only one-third as long as a typical GASS galaxy. We also describe a correlation between local stellar mass density and metallicity, which is valid across all galaxies in our sample. We argue that much of the recent stellar mass growth at the edges of these galaxies can be linked to the accretion or radial transport of relatively pristine gas from beyond the galaxies' stellar disks.