We perform a complete analysis of the augmented version of the Fictitious Domain method (A-BLM/FD), for problems in which the computational mesh is not fitted to the physical domain. Compared to standard Fictitious Domain approaches, the novelty of this method lies in its more suitable extension of the solution within the fictitious region, obtained by additionally requiring the continuity of the gradient across the interface. This strategy ensures a higher global regularity and a consequent optimal convergence of the discrete solution. In this work, we apply the latter method to a Poisson problem with an immersed Dirichlet condition, proving the well-posedness of the continuous and discrete Finite Element formulations. We then move to the approximated formulation, arising from the inexact integration of the coupling terms. For this formulation as well, we prove the well-posedness and compute some case-specific a priori estimates for the quadrature errors. All theoretical results are supported by numerical tests conducted in both two and three-dimensional benchmark problems.