  We consider the planar local patch approximation of $d=2$ fermions at finite
density coupled to a critical boson. In the quenched or Bloch-Nordsieck
approximation, where one takes the limit of fermion flavors $N_f\rightarrow 0$,
the fermion spectral function can be determined {exactly}. We show that one can
obtain this non-perturbative answer thanks to a specific identity of fermionic
two-point functions in the planar local patch approximation. The resulting
spectrum is that of a non-Fermi liquid: quasiparticles are not part of the
exact fermionic excitation spectrum of the theory. Instead one finds continuous
spectral weight with power law scaling excitations as in a $d=1$ dimensional
critical state. Moreover, at low energies there are three such excitations at
three different Fermi surfaces, two with a low energy Green's function $G \sim
(\omega-v_*k)^{-1/2}$ and one with $G \sim |\omega+k|^{-1/3}$.
