In school choice problems, the pursuit of student welfare (efficiency) is constrained by the requirement to respect schools’ priorities (fairness). Among fair matchings, even the welfare maximizing one, the Student-Optimal Stable Matching (SOSM), is inefficient. Moreover, any mechanism that yields welfare gains over the SOSM is manipulable by students. This paper investigates the fairness and incentive properties of efficient mechanisms. We compare matchings using the notion that one matching is less unfair than another if it generates a smaller set of students whose priorities are violated, and we define minimal unfairness accordingly. We show that the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism is minimally unfair within the class of Pareto efficient mechanisms satisfying a simple incentive requirement, top-manipulation-proofness. Moreover, the EADA satisfies a stronger property, upper-manipulation-proofness. Upper-manipulation-proofness is a broad incentive property one can maintain while improving upon the SOSM. Together, these results highlight a sharp frontier in school choice: when the objective is efficiency and improving student welfare over the SOSM, the EADA emerges as a compelling mechanism in both fairness and incentive terms.