  A two-step method for solving planar Laplace problems via rational
approximation is introduced. First complex rational approximations to the
boundary data are determined by AAA approximation, either globally or locally
near each corner or other singularity. The poles of these approximations
outside the problem domain are then collected and used for a global
least-squares fit to the solution. Typical problems are solved in a second of
laptop time to 8-digit accuracy, all the way up to the corners, and the
conjugate harmonic function is also provided. The AAA-least squares combination
also offers a new method for avoiding spurious poles in other rational
approximation problems, and for greatly speeding them up in cases with many
singularities. As a special case, AAA-LS approximation leads to a powerful
method for computing the Hilbert transform or Dirichlet-to-Neumann map.
