In this paper, we consider the inverse problem of reconstructing coated inclusions from overdetermined boundary data. In a first part, a shape identifiability result from a Cauchy data is presented, i.e. with Neumann and Dirichlet boundary as measurements. Then the inverse geometric problem is reduced into a minimzation of a cost-type functional: energy gap tracking functional. Since the boundary conditions are known, the variable of the functional is the shape of the coated inclusions. The shape sensitivity analysis is rigorously performed by means of a La-grangian formulation coupled with paramatrization of the shape. Thus we explicit the gradient of the functional by computing the derivative with respect to the missing shape. The optimization problem is numerically solved by means of gradient-based shape strategy then numerical illustrations are presented.