Simulating incompressible flows around thin shapes (e.g., biological membranes, sails, thin-walled structures in construction) poses significant challenges for conventional boundary fitted CFD due to complex meshing requirements, error-prone volumetric discretization of zero-thickness geometries, and prohibitive remeshing costs in fluid-structure interaction or optimization workflows. Unfitted methods offer a promising alternative, yet existing approaches struggle with immersing thin shapes and robustly enforcing slip boundary conditions while maintaining accuracy and efficiency in complex flow scenarios. We resolve this via dual extension operators within the Shifted Boundary Method (SBM) and a robust Nitsche-based imposition of Navier-slip boundary conditions. The Shifted Boundary Method generally provides robustness and computational efficiency. In contrast to other unfitted methods, it does not require additional degrees of freedom nor the need for cutting operations on the intersected fluid volume elements or integration over intersected elements. Unlike traditional SBM, which uses Taylor expansions, our approach employs least-squares extension operators to extrapolate solution fields between a surrogate boundary and the true boundary of a shape. A thin shape requires a separate solution on both of its sides, therefore the solution might be discontinuous. Crucially, we define a shifted boundary and introduce distinct extension operators for both sides of the true boundary to handle the discontinuous solution space, ensuring robust separation and accurate imposition of boundary conditions. Furthermore, we adapt a Nitsche-based formulation for Navier-slip boundary conditions to the SBM, incorporating tailored penalization and stabilization parameters to capture near-wall physics, enabling efficient high-Reynolds-number simulations. We validate the approach through 2D and 3D numerical experiments. The results demonstrate the proposed method’s potential to effectively handle complex geometries. This paves the way for broader application in fluid-structure interaction, topology optimization, and high-fidelity industrial simulations where geometric complexity and efficiency are paramount.