The design of profile extrusion dies remains a challenging task due to the complex rheological behavior of polymer melts and the geometric intricacies of flow channels. Traditional manual optimization approaches, which rely heavily on human experience, are inefficient and often employ unvalidated heuristics. To address these challenges, we present a deterministic and explainable framework for automatic die design based on adjoint-based shape optimization. This approach enables the computation of sensitivities that directly indicate beneficial modifications to the flow channel geometry. A major difficulty in such optimization processes lies in generating boundary-conforming meshes that evolve consistently with changing geometries. To overcome this issue, we employ a fictitious boundary method that eliminates the need for an explicit surface representation along physical boundaries. A dedicated reconstruction technique is developed to recover accurate sensitivity information at the virtual interface between fluid and solid regions. Furthermore, we introduce a hybrid meshing strategy combining adaptive refinement and mesh motion to maintain mesh quality and sufficient boundary representation throughout the optimization process. The proposed algorithm is demonstrated on both 2D and 3D geometries with varying complexity, including realistic extrusion die flow channels. Several objective functionals relevant to industrial applications, such as pressure drop and flow balance, are considered. The results highlight significant improvements in performance metrics while maintaining numerical robustness. This work showcases the potential of adjoint-based techniques for automated die design in a domain still largely governed by manual trial-and-error procedures, establishing a foundation for data-efficient, sustainable manufacturing workflows using computational rheology.