The determination of spectral properties from first principles can provide powerful connections between microscopic theoretical predictions and experimental data, but requires complex electronic-structure formulations that fall outside the domain of applicability of common approaches, such as density-functional theory. We show here that Koopmans-compliant functionals, constructed to enforce piecewise linearity and the correct discontinuity derivative in energy functionals with respect to fractional occupation-i.e., with respect to charged excitations-provide molecular photoemission spectra and momentum maps of Dyson orbitals that are in excellent agreement with experimental ultraviolet photoemission spectroscopy and orbital tomography data. These results highlight the role of Koopmans-compliant functionals as accurate and inexpensive quasiparticle approximations to the spectral potential.