Iron atoms adsorbed on a Cu(111) surface and buried under polyphenyl dicarbonitrile molecules exhibit strongly spatial anisotropic Kondo features with directionally dependent Kondo temperatures and line shapes, as evidenced by scanning tunneling spectroscopy. First-principles calculations find nearly full polarization for the half-filled Fe 3d(xz) and 3d(yz) orbitals, which therefore can give rise to Kondo screening with the experimentally observed directional dependence and distinct Kondo temperatures. X-ray absorption spectroscopy and X-ray magnetic circular dichroism measurements confirm that the spin in both channels is effectively Kondo-screened. At ideal Fe coverage, these two-orbital Kondo impurities are arranged in a self-assembled honeycomb superlattice.