  Immersing a mobile impurity in a quantum many-body environment can reveal
fundamental properties of the background medium, hence providing a powerful
probe of quantum matter. This approach is particularly intriguing when
considering media with exotic properties, such as strongly-correlated phases
and topological states of matter. In this work, we study the dressing of a
mobile impurity interacting with a chiral mode, as provided by the edge of
topological quantum matter. The resulting ''chiral polaron'' is characterized
by an asymmetric spectral function, which reflects the chirality and group
velocity of the topological edge mode and the drag experienced by the mobile
impurity. We first build our theoretical understanding from an effective
one-dimensional chiral model, which captures the hallmark signatures of the
chiral polaron. We then demonstrate how this simple picture extends to
realistic models of integer and fractional Chern insulator states, by adapting
tensor-network methods to polaron spectroscopy. Injecting mobile impurities on
the edge of topological quantum matter is shown to be a powerful tool to probe
exotic edge properties, particularly suitable for cold-atom experiments.
