The Thouless energy was introduced in the 1970s as a semiclassical energy for electrons diffusing through a finite-sized conductor. It turns out to be an important quantum-mechanical energy scale for many systems ranging from disordered metals to quantum chaos to quantum chromodynamics. In particular, it has been quite successful in describing the properties of Josephson junctions when the barrier is a diffusive normal-state metal. The Thouless energy concept can be generalized to insulating barriers by extracting an energy scale from the two-probe Kubo conductance of a strongly correlated electron system (metallic or insulating) via a generalized definition of the quantum-mechanical level spacing to many-body systems. This energy scale is known to determine the crossover from tunneling to Ohmic (thermally activated) transport in normal tunnel junctions. Here we use it to illustrate how the quasiclassical picture of transport in Josephson junctions is modified as the strongly correlated barrier passes through the Mott transition. Surprisingly, we find the quasiclassical form holds well beyond its putative realm of validity.