We consider the Cauchy problem of recovering both Neumann and Dirichlet data on the inner part of the boundary of an annular domain from measurements of a harmonic function on some part of the outer boundary. Using tools from complex analysis and best approximation in Hardy classes, we present a family of fast data completion algorithms which are shown to provide constructive and robust identification schemes. These are applied to the computation of an impedance or Robin coefficient and are validated by a thorough numerical study.