I grew up in the Boston area and I know these cookies as half moons. The cookie I know is not shortbread or cake but a sort of cakey sugar cookie. The icing is a creamy frosting, not a hard glaze or fondant. The cookies are always inverted and the icing applied to the flat bottom.
My wife and I were repeatedly disappointed with the half moons available in local markets and bakeries so we decided to make our own. After some research, testing, and experimentation we came up with a half moon that matches our childhood memories and is far superior to the local store bought variety.