  We study the physics of a mobile impurity confined in a lattice, moving
within a Bose-Hubbard bath at zero temperature. Within the Quantum Gutzwiller
formalism, we develop a beyond-Fr\"ohlich model of the bath-impurity
interaction. Results for the properties of the polaronic quasiparticle formed
from the dressing of the impurity by quantum fluctuations of the bath are
presented throughout the entire phase diagram, focusing on the quantum phase
transition between the superfluid and Mott insulating phases. Here we find that
the modification of the impurity properties is highly sensitive to the
different universality classes of the transition, providing an unambiguous
probe of correlations and collective modes in a quantum critical many-body
environment.
