Since the breast is one of the most radiosensitive organs, mammography is arguably the area where lowering radiation dose is of the uttermost importance. Phase-based x-ray imaging methods can provide opportunities in this sense, since they do not require x-rays to be stopped in tissue for image contrast to be generated. Therefore, x-ray energy can be considerably increased compared to those usually exploited by conventional mammography. In this article we show how a novel, optimized approach can lead to considerable dose reductions. This was achieved by matching the edge-illumination phase method, which reaches very high angular sensitivity also at high x-ray energies, to an appropriate image processing algorithm and to a virtually noise-free detection technology capable of reaching almost 100% efficiency at the same energies. Importantly, while proof-of-concept was obtained at a synchrotron, the method has potential for a translation to conventional sources.