  We propose a new way to experimentally determine the subleading low-energy
structure constant of doubly-virtual Compton scattering on a proton. Such
empirical determination will reduce the theoretical model error in estimates of
the hadronic correction to the muonic hydrogen Lamb shift. We demonstrate that
the di-lepton forward-backward asymmetry in the $e^- p \to e^- p \, e^- e^+$
process, which can be accessed at electron scattering facilities, yields a
large sensitivity to this so far unknown low-energy constant.
