  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 $e \mu \to e \mu$. I
will discuss the implementation of a Monte Carlo integrator for this process in
the McMule framework arXiv:2212.06481, 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.
