Flat proton energy ($50$ keV – a few MeV) spectra have been measured in-situ at $1$ AU in large solar energetic particle events in the past decades. The diffusive picture of particle acceleration at shocks propagating into media that enforce a uniform spatial diffusion far upstream (several hours at $1$ AU) seems unable to reproduce such measurements. Here we use a method recently developed based on a 1D transport equation accounting for particle acceleration and escape, both allowed at all particle energies, not only the highest energies. The key ingredient of the model is that diffusion is contributed by both self-generated turbulence close to the shock and by pre-existing turbulence far upstream, so that the upstream mean free path gradually increases out to a saturation in the pristine solar wind turbulence. The steady-state spectra agree very well with in-situ measurements, confirming that escape into a medium with mean free path spatial variation is a crucial ingredient in forging the energy spectra in the proximity of shocks.