This paper proposes a novel distributed solution for the operation of large populations of thermostatically controlled loads (TCLs) providing frequency support. A game-theory framework is adopted, modeling the TCLs as price-responsive rational agents that schedule their energy consumption and allocate frequency response provision in order to minimize their operational costs. The novelty of this work lies in the use of mean field games to abstract the complex interactions of large numbers of TCLs with the grid and in the introduction of an innovative market structure, envisioning distinct price signals for electricity and response. Differently from previous approaches, such prices are not designed ad hoc but are derived instead from an underlying system scheduling model.