  I present the computation of the two-loop amplitudes for the scattering of a
lepton pair with an off-shell and an on-shell photon in massless QED. We apply
modern techniques developed to tackle QCD amplitudes with many scales: we
express the Feynman integrals in terms of a basis of special functions, and
reconstruct the amplitudes from numerical finite-field evaluations. Our results
complete the amplitude-level ingredients for the N3LO predictions of
electron-muon scattering needed to meet the precision target of the future
MUonE experiment.
