We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂Ω. Assuming Dirichlet and Neumann data available on Γ ⊂ ∂Ω to be real-valued functions in W 1/2,2 (Γ) and L 2 (Γ) classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 bound of the solution on ∂Ω \ Γ as a regularizing parame- ter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints on the solution which, on practical side, can help incorporating outlying boundary measurements without changing the boundary into a less regular one.