The description of hadronic structure in terms of quark and gluon degrees of freedom is an open subject in physics. Great efforts are being devoted to it on both the theoretical and experimental sides. Triggered by existing plans to build new experimental facilities, and the need to properly interpret the data that are to come, interest into the zoology of parton distribution functions is increasing. Among them, GPDs, which are known to parametrize the soft-physics taking place in DVCS, are expected to play a central role drawing three-dimensional images of hadrons. In this work we focus on the study of pions, which, as Nambu-Goldstone bosons of QCD chiral symmetry breaking, provide one of the clearest windows onto the phenomenon of emergent hadronic mass (EHM). Herein we present a novel class of pion off-forward parton distributions (GPDs) which, relying on the covariant-extension, fulfill all of the theoretical constraints required by QCD. We analyse the effect of scale-evolution, showing that gluon content plays a dominant role into the description of pion's response to Compton scattering (CFFs). Finally, we exploit our models to obtain predictions for DVCS on pions, which is to be probed through the Sullivan process at the foreseen EIC. We show that a measurable asymmetry arise on that channel, therefore pushing optimism about probing pion's structure at future electron-ion colliders.