This work concerns the dynamics of a two‐dimensional (surface) fluid membrane which resists bending in a manner reminiscent of elastic shells. Such bending stiffness can arise from the molecular structure of the membrane, the molecules being treated as “directors” oriented normal to the surface. In this paper we unify the hydrodynamics of a two‐dimensional viscoelastic fluid with the mechanics of an elastic shell in the spirit of a Cosserat continuum. The kinematics of the evolving surface geometry is developed, along with the dynamics of this surface phase, in the time‐dependent, non‐Euclidean metric space of the surface itself. Considerations of linear‐ and angular‐momentum balance lead to an asymmetric surface stress tensor as well as a moment (or couple‐stress) tensor for which a variety of constitutive relations are considered. The antisymmetric component of stress, along with a transverse shear force, couples the membrane bending motion to the surface flow vorticity. The relevance of this theory to the dynamics of coated droplets in general, and to biological cell membranes in particular, is