We study the simultaneous recovery of the boundary and coefficient of the Robin boundary condition for the Laplace equation from a pair of solution measurements on another part of the boundary. We derive the variational derivatives of the data-fitting objective functional with respect to the Robin boundary and coefficient, which are then used to device a nonlinear conjugate gradient iterative scheme for the numerical recovery of both the Robin boundary and coefficient together. Numerical examples are presented to illustrate the effectiveness of the recovery algorithms.