A mechanical model for anisotropic curved interfaces, applicable to thermodynamically closed surfactant-laden liquid-liquid crystal interfaces is developed. The model takes into account the mechanical effects due to surface bending and surface tilting (anchoring) and incorporates liquid crystal anisotropy into classical fluid membrane mechanics. In the absence of the aligned liquid crystal, the model converges to the fluid membrane mechanical model, and in the absence of surfactant, it converges to the nematic interface mechanical model. Use of the well-known Helfrich-Rapini-Papoular surface energies leads to the Laplace equation for anisotropic curved interfaces, whose material limits are the vesicle shape equation and the liquid crystal Herring equation. Applications of the model to shape selection in liquid drops embedded in aligned nematic liquid crystals illustrates the competition between anchoring and bending and shows how anisotropic surface tension distorts the droplet and how bending tends to restore the spherical shape. This theory presented in this article shows that the interaction of interfacial anchoring and bending creates new regimes in classical fluid membrane mechanics.