We present an extension of the shifted boundary method to simulate partial differential equations with moving internal interfaces. The objective is to apply the method to phase change problems modelled by the Stefan equation : the parabolic heat equation with a discontinuity separating the two phases, moving at a speed given by the normal flux jump. To obtain an accurate prediction of the temperature field on both sides of the discontinuity, and of the position of the discontinuity itself, we propose an enhanced variant of the shifted boundary method. This method is based on an enriched mixed form proposed by some of the present authors, which allows a uniform second order accuracy in space and time. Stabilization terms are added on the whole domain to ensure convergence. The specificities generated by the interface displacement are handled by a particular construction, with a double nodal points structure at the surrogate interface. Several recovering techniques based on Taylor developments from the interface to its surrogate, and a least square minimization method for node initialization are performed to recover the missing values generated by the moving interface and the phase structure of the model.