In a quasi-two-dimensional electron system with nonzero layer thickness, a parallel magnetic field can couple to the out-of-plane electron motion and lead to a severe distortion and eventual splitting of the Fermi contour. Here we directly and quantitatively probe this evolution through commensurability and Shubnikov-de Haas measurements on electrons confined to a 40-nm-wide GaAs (001) quantum well. We are able to observe the Fermi contour splitting phenomenon, in good agreement with the results of semiclassical calculations. Experimentally, we also observe intriguing features, suggesting magnetic-breakdown-type behavior when the Fermi contour splits.