We investigate a model of R²-gravity with a non-minimally coupled scalar field that gives rise to two-stage inflation with a break, that is, with an intermediate stage where inflation momentarily halts. We find that the power spectrum of the primordial curvature perturbation is significantly enhanced at the break scale, which can account for the primordial black hole (PBH) formation, without affecting the CMB constraint on large scales. The behavior of the curvature perturbation is carefully analyzed and we find a few notable new features in the spectrum. In particular, we find that the k³ growth of the spectrum of toward the end of the first stage of inflation. We argue that this is a universal feature common to all two-stage models where the field dominating the second stage is heavy during the first stage. By appropriately tuning the model parameters, we find that our model can realize the scenario of PBHs as the cold dark matter of the Universe. We also find that we can choose the parameters so that the spectrum of the induced gravitational waves from the enhanced curvature perturbation fits the NANOGrav-15yr data of pulsar timing array observation.