Open colloidal lattices possessing omnidirectional photonic bandgaps in the visible or near-visible regime are attractive optical materials the realization of which has remained elusive. We report the use of an inverse design strategy termed landscape engineering that rationally sculpts the free energy self-assembly landscape using evolutionary algorithms to discover anisotropic patchy colloids capable of spontaneously assembling pyrochlore and cubic diamond lattices possessing complete photonic bandgaps. We validate the designs in computer simulations to demonstrate the defect-free formation of these lattices via a two-stage hierarchical assembly mechanism. Our approach demonstrates a principled strategy for the inverse design of self-assembling colloids for the bottom-up fabrication of desired crystal lattices.