  A major driver of quantum-simulator technology is the prospect of probing
high-energy phenomena in synthetic quantum matter setups at a high level of
control and tunability. Here, we propose an experimentally feasible realization
of a large-scale $2+1$D $\mathrm{U}(1)$ gauge theory with dynamical matter and
gauge fields in a cold-atom quantum simulator with spinless bosons. We present
the full mapping of the corresponding Gauss's law onto the bosonic
computational basis. We then show that the target gauge theory can be
faithfully realized and stabilized by an emergent gauge protection term in a
two-dimensional single-species Bose--Hubbard optical Lieb superlattice with two
spatial periods along either direction, thereby requiring only moderate
experimental resources already available in current cold-atom setups. Using
infinite matrix product states, we calculate numerical benchmarks for adiabatic
sweeps and global quench dynamics that further confirm the fidelity of the
mapping. Our work brings quantum simulators of gauge theories a significant
step forward in terms of investigating particle physics in higher spatial
dimensions, and is readily implementable in existing cold-atom platforms.
