Using a state-of-the art combination of density functional theory and impurity solver techniques, we present a complete and parameter-free picture of the Kondo effect in the high-spin (S = 3/2) coordination complex known as manganese phthalocyanine adsorbed on the Pb(111) surface. We calculate the correlated electronic structure and corresponding tunnel spectrum and find an asymmetric Kondo resonance, as recently observed in experiments. Contrary to previous claims, the Kondo resonance stems from only one of three possible Kondo channels with origin in the Mn 3d orbitals, its peculiar asymmetric shape arising from the modulation of the hybridization due to a strong coupling to the organic ligand. The spectral signature of the second Kondo channel is strongly suppressed as the screening occurs via the formation of a many-body singlet with the organic part of the molecule. Finally, a spin-1/2 in the 3d shell remains completely unscreened due to the lack of hybridization of the corresponding orbital with the substrate, hence leading to a spin-3/2 underscreened Kondo effect.