We propose a minimal model to study the real-time dynamics of a ${{\mathbb{Z}}}_{2}$ lattice gauge theory (LGT) coupled to fermionic matter in a cold atom quantum simulator setup. We show that dynamical correlators of the gauge fields can be measured in experiments studying the time-evolution of two pairs of impurities, and suggest the protocol for implementing the model in cold atom experiments. Further, we discuss a number of unexpected features found in the integrable limit of the model, as well as its extensions to a non-integrable case. A potential experimental implementation of our model in the latter regime would allow one to simulate strongly-interacting LGT beyond current capabilities of classical computers.