Unfitted meshes are increasingly important in computational mechanics, enabling efficient handling of complex geometries, such as trimmed domains, without requiring boundary-conforming discretizations. While unfitted boundary descriptions offer significant advantages, they also introduce the challenge of handling boundary conditions, which are often addressed using weak enforcement techniques. This work presents a physics-agnostic, non-intrusive, iterative method for imposing strong Dirichlet boundary conditions on unfitted meshes, effectively addressing challenges like ill-posed systems and small cut-cell instabilities, which are commonly encountered in numerical methods with unfitted boundaries, e.g. [2]. The method reformulates boundary condition enforcement as an L2-norm error minimization problem, augmented by a stabilization term to approximate the normal gradient within trimmed elements. Gradient approximation within trimmed elements leverages the calculated gradient from neighboring untrimmed elements. Techniques such as Radial Basis Function (RBF) interpolation and Moving Least Squares (MLS) are compared to evaluate their relative advantages and disadvantages in terms of stability and accuracy. Crucially, this approach eliminates the need for fine-tuning parameters, such as penalty factors used in Penalty-based weak imposition methods. Importantly, the weak form of the governing equations remains unaltered, preserving the problem's intrinsic properties. Validated through convergence studies for the Poisson problem using both Finite Elements (FEM) and IGA [1] discretizations (B-Splines), the method demonstrates optimal L2-norm error convergence and improved system conditioning. In addition, a comparison study of this new method with IBRA [2] was conducted. In this work, it will also be demonstrated that for explicit dynamic simulations where time steps involved are usually very small and the gradient solution changes only slightly between them, good accuracy is obtained without iterating.