We measure angle differential cross sections (DCS) in Li+ + Na → Li + Na+ electron transfer collisions in the 2.7–24 keV energy range. We do this with a newly constructed apparatus which combines the experimental technique of cold target recoil ion momentum spectroscopy with a laser-cooled target. This setup yields a momentum resolution of 0.12 au, an order of magnitude better angular resolution than previous measurements on this system. This enables us to clearly resolve Fraunhofer-type diffraction patterns in the angle DCS. In particular, the angular width of the ring structure is given by the ratio of the de Broglie wavelength λdB = 150 fm at a velocity v = 0.20 au and the effective atomic diameter for electron capture 2R = 20 au. Parallel AO and MO semiclassical coupled-channel calculations of the Na(3s, 3p) → Li(2s, 2p) state-to-state collision amplitudes have been performed, and quantum scattering amplitudes are derived by the eikonal method. The resulting angle-differential electron transfer cross sections and their diffraction patterns agree with the experimental level-to-level results over most scattering angles in the energy range.