High-mobility graphene hosting massless charge carriers with linear dispersion provides a promising platform for electron optics phenomena. Inspired by the physics of dielectric optical microcavities where the photon emission characteristics can be efficiently tuned via the cavity shape, we study corresponding mechanisms for trapped Dirac fermionic resonant states in deformed microdisk graphene billiards and directed emission from those. In such graphene devices a back-gate voltage provides an additional tunable parameter to mimic different effective refractive indices and thereby the corresponding Fresnel laws at the boundaries. Moreover, cavities based on single-layer and double-layer graphene exhibit Klein- and anti-Klein tunneling, respectively, leading to distinct differences with respect to dwell times and resulting emission profiles of the cavity states. Moreover, we find a variety of different emission characteristics depending on the position of the source where charge carriers are fed into the cavities. Combining quantum mechanical simulations with optical ray tracing and a corresponding phase-space analysis, we demonstrate strong confinement of the emitted charge carriersin the midfield of single-layer graphene systems and can relate this to a lensing effect. For bilayer graphene, trapping of the resonant states is more efficient and the emission characteristics do depend less on the source position.