A program devoted to perform the first in-vivo monochromatic breast computed tomography (BCT) is ongoing at the Elettra Synchrotron Facility. Since the synchrotron radiation provides high energy resolution and spatial coherence, phase-contrast (PhC) imaging techniques can be used. The latest high resolution BCT acquisitions of breast specimens, obtained with the propagation-based PhC approach, are herein presented as part of a wider framework, devoted to the optimization of acquisition and reconstruction parameters towards the clinical exam. Images are acquired with a state-of-the-art dead-time-free single-photon-counting CdTe detector with a 60 µm pixel size. The samples are imaged at 32 and 38 keV in continuous rotating mode, delivering 5-20 mGy of mean glandular dose (MGD). Contrast-to-noise ratio (CNR) and spatial resolution performances are evaluated for both absorption and phase-retrieved images considering tumor/adipose tissue interfaces. We discuss two different phase-retrieval approaches, showing that a remarkable CNR increase (from 0.5 to 3.6) can be obtained without a significant loss in spatial resolution. It is shown that, even if the non-phase-retrieved image has a poorer CNR, it is useful for evaluating the spiculation of a microcalcification: in this context, absorption and phase-retrieved images have to be regarded as complementary information. Furthermore, the first full volume acquisition of a mastectomy, with a 9 cm diameter and 3 cm height, is reported. This investigation on surgical specimens indicates that monochromatic BCT with synchrotron radiation in terms of CNR, spatial resolution, scan duration and scan volume is feasible.