Thermal transport in metal-oxide-semiconductor field effect transistors (MOSFETs) due to electron-phonon scattering is simulated using phonon generation rates obtained from an electron Monte Carlo device simulation. The device simulation accounts for a full band description of both electrons and phonons considering 22 types of electron-phonon scattering events. Detailed profiles of phonon emission/absorption rates in the physical and momentum spaces are generated and are used in a MOSFET thermal transport simulation with a recently-developed anisotropic relaxation time model based on the Boltzmann transport equation (BTE). Comparisons with a Fourier conduction model reveal that the anisotropic heat conduction model predicts higher maximum temperatures because it accounts for the bottlenecks in phonon scattering pathways. Heat fluxes leaving the boundaries associated with different phonon polarizations and frequencies are also examined to reveal the main modes responsible for transport. It is found that though the majority of the heat generation is in the optical modes, the heat generated in the acoustic modes is not negligible. The modes primarily responsible for the transport of heat are found to be medium-to-high frequency acoustic phonon modes.