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Copolymer nanovesicles are used extensively in chemical processes and biomedical applications in which they are subjected to dynamic flow environments. Flow-induced vesicle deformation, fragmentation, and reorganization modify the energetic (e.g., polymer–solvent interfacial area) and entropic (e.g., copolymer chain configuration) contributions to the solution free energy. Equilibration of a deformed morphology by flow cessation could reorganize the system into a self-assembled state, which is different from the parent structure through a local free energy minimization pathway. We perform nonequilibrium molecular dynamics simulations to investigate morphology evolution in uniform shear flow of a unilamellar nanovesicle formed by the self-assembly of amphiphilic triblock copolymers in an aqueous solution. Flow strength is characterized by the Weissenberg number Wi, defined as the ratio of the time scale of vesicle shape fluctuations to the inverse shear rate. For Wi< 10, a spherical vesicle deforms into a flow-aligned ellipsoidal bilayer executing tank-treading motion. For Wi> 10, pronounced variations in bilayer thickness and polymer extension manifest along the contour of the elongated vesicle, which breaks up into lamellar fragments. Below a critical strain, the deformed vesicle upon flow cessation returns to its initial spherical morphology. However, for larger strains, structure reorganization after flow stoppage results in the formation of a Novel Equilibrated Shear-Induced Structure (NoESIS) in which two vesicles are connected by a dynamic molecular bridge that can accommodate additional layers of copolymers, leading to a reduction in the polymer–solvent interface area. Mechanisms of morphology hysteresis are explored via an analysis of the thermodynamic markers.