Unlike the electrical conductance that can be widely modulated within the same material even in deep-subwavelength devices, tuning the thermal conductance within a single material system or nanostructure is extremely challenging and requires a large-scale device. This prohibits the realization of robust on/off states in switching the flow of thermal currents. Here, we present the theory of a thermal switch based on resonant coupling of three photonic resonators, in analogy to the field-effect electronic transistor composed of a source, a gate, and a drain. As a material platform, we capitalize on the extreme tunability and low-loss resonances observed in the dielectric function of monolayer hexagonal boron nitride (h-BN) under controlled strain. We derive the dielectric function of h-BN from first principles, including the phonon-polariton line widths computed by considering phonon-isotope and anharmonic phonon-phonon scattering. Subsequently, we propose a strain-controlled h-BN–based thermal switch that modulates the thermal conductance by more than an order of magnitude, corresponding to a contrast ratio in the thermal conductance of 98%, in a deep-subwavelength nanostructure.