Heat transfer at the nanoscale remains largely unexplored compared to that in bulk materials, particularly phonon tunneling across extremely narrow nanogaps. The underlying mechanisms of this phenomenon are not well understood due to experimental difficulties. As a novel approach for measuring heat transfer in nanogaps, we have developed an on-chip adiabatic mechanically controllable break junction system based on a suspended membrane device. The electrical and thermal conductance were measured simultaneously across the gold nanogap, with the spacing controlled by a built-in actuator. Our results indicate anomalous thermal conductance at gap distances of a few nanometers, strongly suggesting phonon-mediated heat transfer in these extremely confined spaces.