  One of the challenging goals in the studies of many-body physics with
ultracold atoms is the creation of a topological $p_{x} + ip_{y}$ superfluid
for identical fermions in two dimensions (2D). The expectations of reaching the
critical temperature $T_c$ through p-wave Feshbach resonance in spin-polarized
fermionic gases have soon faded away because on approaching the resonance, the
system becomes unstable due to inelastic-collision processes. Here, we consider
an alternative scenario in which a single-component degenerate gas of fermions
in 2D is paired via phonon-mediated interactions provided by a 3D BEC
background. Within the weak-coupling regime, we calculate the critical
temperature $T_c$ for the fermionic pair formation, using Bethe-Salpeter
formalism, and show that it is significantly boosted by higher-order
diagramatic terms, such as phonon dressing and vertex corrections. We describe
in detail an experimental scheme to implement our proposal, and show that the
long-sought p-wave superfluid is at reach with state-of-the-art experiments.
