This is the first in a series of four papers developing a scattering theory for harmonic functions/one-forms on Riemann surfaces. In this paper we prove the following. Let R be a compact Riemann surface split into two surfaces Sigma(1) and Sigma(2) by a complex of quasicircles. Given a harmonic function with L-2 derivatives on one of the pieces Sigma(1), there is a unique harmonic function with L-2 derivatives on the other piece Sigma(2) with the same boundary values as the original function in a certain conformally invariant non-tangential sense. We call the new harmonic function the overfare of the original function. This overfare map is well-defined and bounded with respect to Dirichlet semi-norm provided that Sigma(1) is connected. For Weil-Petersson quasicircles, it is bounded with respect to the Sobolev H-1-norm.