We demonstrate a nanostructure composed of partially etched annular trenches in a suspended GaAs membrane, designed for efficient and moderately broadband (≈5 nm) emission extraction from single InAs quantum dots. Simulations indicate that a dipole embedded in the nanostructure center radiates upward into free space with a nearly Gaussian far field, allowing a collection efficiency >80% with a high numerical aperture (NA = 0.7) optic and with ≈12× Purcell radiative rate enhancement. Fabricated devices exhibit a ≈10% photon collection efficiency with a NA = 0.42 objective, a 20× improvement over quantum dots in unpatterned GaAs. A fourfold exciton lifetime reduction indicates moderate Purcell enhancement.