Breast cancer is the most common malignancy in women. Unfortunately, even though screening programs have helped to increase survival rates, the number of false positives and false negatives remains high. phase-contrast X-ray CT is a promising imaging technique which could improve breast cancer diagnosis by combining the high three-dimensional resolution of conventional CT with higher soft-tissue contrast. Grating Interferometry CT (GI-CT) arguably has the highest chance to make the transition to clinical practice. Unfortunately though, obtaining high-quality images is challenging. Grating fabrication defects and photon starvation lead to high noise amplitudes in the measured data. Moreover, the highly ill-conditioned differential nature of the GI-CT forward operator renders the inversion from corrupted data even more cumbersome. In this article we report on a novel regularized iterative reconstruction algorithm with a powerful data-driven regularization strategy to tackle this challenging inverse problem. In particular, we present an algorithm that combines the L-BFGS optimization scheme with a Plug-and-Play denoiser parameterized by a deep neural network and empirically show that the proposed method achieves high quality images, both on simulated data as well as on real measurements.