We study a class of degenerate fully nonlinear elliptic equations consisting of a pure equation and an associated transmission-type free boundary problem. We first propose a regularization of the pure equation and derive a numerical method for the regularized problem. The regularization plays a crucial role in ensuring the monotonicity of the numerical scheme. We prove that the method is monotone, consistent, and stable. As a consequence, the Barles–Souganidis framework guarantees the convergence of the numerical approximation. Once the numerical treatment of the pure equation is established, we show how a similar approach can be extended to the free boundary transmission problem. Finally, we present numerical experiments that support the theoretical results.