  Charge carrier doping of a Mott insulator is known to give rise to a wide
variety of exotic emergent states, from high-temperature superconductivity to
various charge, spin, and orbital orders. The physics underpinning their
evolution is, however, poorly understood. A major challenge is the chemical
complexity associated with traditional routes to the addition or removal of
carriers. Here, we study the Mott insulating CrO$_2$ layer of the delafossite
oxide PdCrO$_2$, where an intrinsic polar catastrophe provides a clean route to
induce substantial doping of the surface layer. Despite this, from scanning
tunneling microscopy and angle-resolved photoemission, we find that the surface
retains an insulating character, but with a modified electronic structure and
the development of a short-range ordered state with a distinct
$(\sqrt{7}\times\sqrt{7})\mathrm{R}\pm 19.1^\circ$ periodicity. From density
functional theory, we demonstrate how this reflects the formation of an
intricate charge disproportionation that results in an insulating ground state
of the surface layer that is disparate from the hidden Mott insulator found in
the bulk. By applying voltage pulses to the surface layer, we induce
substantial local modifications to this state, which we find relax on a time
scale of tens of minutes, pointing to a glassy nature of the
charge-disproportionated insulator realised here.
