The paper describes a new method of electronic refrigeration based on resonant Fowler-Nordheim emission. In this method, a bulk emitter is covered with a-few-nm-thick film of a widegap semiconductor, creating an intermediate step between electron energies in the emitter and in vacuum. An external electric field tilts this potential profile, forming a quantum well at the semiconductor-vacuum boundary. Quantization of electron motion in this well creates a system of ID levels for the motion perpendicular to the surface (subbands for the total electron energy). Alignment of its lowest subband with the energy of the hottest electrons of the emitter (a few kBT above the Fermi level) leads to a resonant, selective emission of these electrons, providing emitter cooling. On the other hand, the coldest electrons can only tunnel out through a thicker, composite barrier, and the associated heating may be low. Calculations show that cooling power well beyond 100 W/cm2 (at 300 K), as well as temperatures down to at least 100 K, may be achieved using this effect in a system using electron energy recovery.