If sub-GeV dark matter (DM) reaches thermal equilibrium with the Standard Model (SM) in the early universe, its large thermal number density must be depleted by annihilation into the SM. If DM is SM neutral, this annihilation requires an additional light particle to mediate dark/visible interactions. The minimum annihilation rate to evade overclosure sets a clear benchmark for future experimental efforts to discover or falsify this broad class of models. For virtually the entire sub-GeV mass range, this benchmark can be decisively probed by with a dedicated experiment that exploits the distinctive kinematics of DM production in electron-nucleus collisions. In this setup, individual electrons are fired through a thin target adjacent to a tracker and calorimeter. If DM particles are produced as the electron passes through the target, they carry away a large fraction of the incident electron's energy. Surprisingly, with suitable trigger and kinematic requirements, an experiment of this sort can cover nearly all the parameter space consistent with a thermal relic abundance, thereby testing all light-mediator models that have ever achieved thermal equilibrium with the SM -- including scenarios that evade existing CMB and direct detection constraints.