In shape optimization, the definition of the geometry depends on the method employed.The most common method (SIMP) utilizes a regular mesh with the boundary defined by thresholding the density defined at element. Hence, the boundary is mesh dependent and not precisely defined. One way to address this limitation is to render the boundary completely independent of the mesh. Various methods have been developed for this purpose, including CutFEM, cg-FEM, FiniteCells and fg-FEM, among others.\par In this work we use the CutFEM to solve PDE on moving boundary domain, without remeshing and preserving a good system's conditionning. The level-set method is used to define the boundary, and the domain evolution is governed by Hamilton-Jacobi (HJ) equation. The velocity field, previously calculated, corresponds to the descent direction of the shape derivative of the cost function. However, HJ's solutions do not inherently preserve the sign distance property, which is essential to calculate the normal vector at the boundary. To address this issue, we propose a new technique to reinitialize the level set function during shape optimization in the CutFEM framework, it is inspired by the DRLSE reinitialization method. We will present the first results of our algorithm implementation with CutFEM implementation in FEniCSX. First, we will demonstrate that our method yields excellent results for the minimisation of compliance under constrained volume in comparison to the fictitious material methods. Emphasis will be placed on the benefits of the method with a coarse mesh and on the efficiency of the reinitialization method in the context of shape optimization. Secondly, non-auto-adjoint problems will be introduced, such as minimizing the structure's volume with a Von Mises global criteria constraint. All implementations are carried our in FEniCSX benefiting from automatic differentiation.