A recently proposed experiment, MUonE, aims to extract the hadronic vacuum polarisation contribution to the muon g − 2 from muon-electron scattering at low energy. The extrapolation requires that both experimental and theoretical uncertainties do not exceed 10 ppm. This corresponds, at least, to next-to-next-to-leading-order (NNLO) QED corrections to eμ → eμ. I will discuss the implementation of a Monte Carlo integrator for this process in the McMule framework [1], which provides infrared-safe differential results at said order in QED. An approximation of the MUonE setup provides some phenomenological results and sheds light on the need for beyond-NNLO corrections, which are currently under study within McMule.