  We develop a non-Gaussian variational approach that enables us to study both
equilibrium and far-from-equilibrium physics of the two-dimensional Fermi
polaron. This method provides an unbiased analysis of the polaron-to-molecule
phase transition without relying on truncations in the total number of
particle-hole excitations. Our results -- which include the ground state energy
and quasiparticle residue -- are in qualitative agreement with the known Monte
Carlo calculations. The main advantage of the non-Gaussian states compared to
conventional numerical methods is that they enable us to explore long-time
polaron evolution and, in particular, study various spectral properties
accessible to both solid-state and ultracold atom experiments. We design two
types of radiofrequency spectroscopies to measure polaronic and molecular
spectral functions. Depending on the parameter regime, we find that these
spectral functions and fermionic density profiles near the impurity display
either long-lived oscillations between the repulsive and attractive polaron
branches or exhibit fast relaxational dynamics to the molecular state.
