We solve first-kind Fredholm boundary integral equations arising from Helmholtz and Laplace problems on bounded, smooth screens in three dimensions with either Dirichlet or Neumann conditions. The proposed Galerkin-Bubnov methods take as discretization elements pushed-forward weighted azimuthal projections of standard spherical harmonics onto the unit disk. By exactly depicting edge singular behaviors we show that these spectral or high-order bases yield super-Algebraic error convergence in the corresponding energy norms whenever the screen is an analytic deformation of the unit disk. Moreover, we provide a fully discrete analysis of the method, including quadrature rules, based on analytic extensions of the spectral basis to complex neighborhoods. Finally, we include numerical experiments to support our claims as well as appendices with computational details for treating the associated singular integrals.