Simultaneously controlling both the spectral and angular emission of thermal photons can qualitatively change the nature of thermal radiation, and offers a great potential to improve a broad range of applications, including infrared light sources and thermophotovoltaic (TPV) conversion of waste heat to electricity. For TPV in particular, frequency-selective emission is necessary for spectral matching with a photovoltaic converter, while directional emission is needed to maximize the fraction of emission reaching the receiver at large separation distances. This can allow the photovoltaics to be moved outside vacuum encapsulation. In this work, we demonstrate both directionally and spectrally-selective thermal emission for p-polarization, using a combination of an epsilon-near-zero (ENZ) thin film backed by a metal reflector, a high contrast grating, and an omnidirectional mirror. Gallium-doped zinc oxide is selected as an ENZ material, with cross-over frequency in the near-infrared. The proposed structure relies on coupling guided modes (instead of plasmonic modes) to the ENZ thin film using the high contrast grating. The angular width is thus controlled by the choice of grating period. Other off-directional modes are then filtered out using the omnidirectional mirror, thus enhancing frequency selectivity. Our emitter design maintains both a high view factor and high frequency selectivity, leading to a factor of 8.85 enhancement over a typical blackbody emitter, through a combination of a 22.26% increase in view factor and a 6.88x enhancement in frequency selectivity. This calculation assumes a PV converter five widths away from the same width emitter in 2D at 1573 K.