We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal \( C^{1,1} \)-regularity estimate for \( L^p \)-strong solutions at points where the free and fixed boundaries intersect. A key novelty is that no convexity or concavity assumptions are imposed on the fully nonlinear operator governing the system. Our analysis derives BMO estimates in a universal neighbourhood of the fixed boundary. It relies solely on a differentiability assumption. Once those estimates are available, applying by now standard methods yields the optimal regularity.