  Recent realization of well-controlled quantum simulators of the
triangular-lattice Fermi-Hubbard model, including the triangular optical
lattices loaded with ultracold Fermions and the heterostructures of the
transition-metal dichalcogenides, as well as the more advanced techniques to
probe them, pave the way for studying frustrated Fermi-Hubbard physics. Here,
we theoretically predict asymmetric photoemission and tunneling spectroscopies
for a lightly hole-doped and electron-doped triangular Mott antiferromagnet,
and reveal two distinct types of magnetic polarons: a \emph{lightly}
renormalized quasiparticle with the same momentum as the spin background and a
\emph{heavily} renormalized quasiparticle with a shifted momentum and a nearly
flat band, using both analytical and unbiased numerical methods. We propose
these theoretical findings to be verified in frustrated optical lattices and
Moir\'e superlattices by probing various observables including the spectral
function, the density of states, the energy dispersion and the quasiparticle
weight. Moreover, we reveal the asymmetric response of the spin background
against charge doping, demonstrating that the interplay between the local spin
and charge degrees of freedom plays a vital role in doped triangular Mott
antiferromagnets.
