We review our results on a new class of $2 \to 3$ exclusive processes, as a probe of both chiral-even and chiral-odd quark GPDs. We consider the exclusive photoproduction of a photon-meson pair, in the kinematics where the pair has a large invariant mass, described in the collinear factorization framework. We cover the whole kinematical range from medium energies in fixed target experiments to very large energies of colliders, by considering the experimental conditions of JLab 12-GeV, COMPASS, future EIC and LHC (in ultra-peripheral collisions) cases. Our analysis covers neutral and charged rho-mesons, as well as charged pions. The case of the rho-meson, depending on its polarization, provides access to either chiral-even or chiral-odd GPDs, at leading twist. We find that the order of magnitude of the obtained cross sections are sufficiently large for a dedicated experimental analysis to be performed, especially at JLab. Furthermore, we compute the linear photon beam polarization asymmetry, which we find to be sizeable, in the case of a longitudinally polarized $\rho$-meson or of a charged pion. These predictions are obtained for both asymptotic distribution amplitude (DA) and holographic DA.