In this paper, an innovative microwave imaging (MI) approach for breast tumor diagnosis is proposed that employs a differential formulation of the inverse scattering problem (ISP) at hand to exploit arbitrary-fidelity priors on the inhomogeneous reference/healthy tissues. The quantitative imaging of the unknown tumor is then rephrased into a global optimization problem, which is efficiently solved with an ad-hoc physics-driven artificial intelligence (AI) strategy inspired by the concepts and guidelines of the System-by-Design (SbD) paradigm. The effectiveness, the robustness, the reliability, and the efficiency of the proposed method are assessed against both synthetic and experimental data.