Notched structural components are widely employed in engineering to meet specific functional and structural demands. However, the presence of notches inherently induces stress concentrations, which can significantly compromise fatigue life and fracture toughness. This study presents an analytical derivation of a closed-form solution for the optimal profile of a deep-notch, utilizing complex analysis combined with the principle of minimum potential energy. Considerations provide additional physical insight into the famous cycloidal curve proving that it can serve as an optimal profile for the bottom of a deep-notch in the elastic domain, providing minimum stress concentration in the torsion problem. To validate the closed-form solution, high-fidelity numerical simulations are conducted using an advanced boundary element method (BEM), recognized for its exceptional accuracy in resolving stress fields near singularities or high gradients. Beyond validation, the BEM serves as a versatile computational platform that overcomes limitations of the analytical model, enabling evaluation of optimal notch geometries under complex configurations and loading conditions. Notably, a counterintuitive phenomenon is revealed within linear elastic torsion, a vanishing radius of curvature at the notch root does not necessarily induce extreme stress concentrations. These findings provide further insight into notch-induced stress amplification and offer new insights into the mechanics of geometric discontinuities. The results advance the fundamental understanding of stress concentration in notched components and provide practical implications for designing fatigue-resistant and damage-tolerant engineering structures.