The GeV-scale gamma-ray excess observed from the region surrounding the Galactic Center has been interpreted as either the products of annihilating dark matter particles, or as the emission from a large population of faint and centrally located millisecond pulsars. If pulsars are responsible for this signal, then they should also produce detectable levels of TeV-scale emission. In this study, we employ a template-based analysis of simulated data in an effort to assess the ability of the Cherenkov Telescope Array to detect or constrain the presence of this emission, providing a new and powerful means of testing whether millisecond pulsars are responsible for the observed excess. We find that after even a relatively brief observation of the inner galaxy, the Cherenkov Telescope Array will be able to definitively detect this TeV-scale emission, or rule out pulsars as the source of the Galactic Center gamma-ray excess.