Mammography and breast CT are important tools for breast cancer screening and diagnosis. Current implementations are limited by scattered radiation and/or spatial resolution. In this work, we propose and develop a slot scan-based system to be used in both mammography and CT mode that can limit scatter and collect sparse CT data for improved image quality at low radiation exposures. Monte Carlo simulations of an anthropomorphic breast phantom show a factor of 10 reduction in scattering amplitude with our slot scan-based system compared to that of a full-field detector mammography system (area mode). Similarly, slot-scan improved the MTF (particularly the low-frequency response) compared to an area detector. Investigation of sparse CT sampling with doubly sparse acquisition data return better quality reconstruction, for which our slot-scanning system is capable, over angle-only projection. Thus, a system with the combined ability for slot-scanning mammography and slot-scanning breast CT has the potential to deliver improved dose-efficient imaging performance and become viable breast cancer screening and diagnostic tools.