Layers of subwavelength metal nanostructures that support localized surface plasmon resonances are of broad interest in applied nanotechnology, for example, in optical sensor development and solar energy harvesting devices. We measured specular reflection spectra as a function of incidence angle for two-dimensional layers of gold nanodisks on glass and found highly asymmetric line-shapes and a spectral red-shift of up to 0.2 eV, or 10% of the plasmon resonance energy, as the angle changed from normal toward grazing incidence. This dramatic angular dispersion is the result of a tunable Fano interference between the spectrally narrow plasmon emission and a "white" continuum caused by the interface reflection. The data are found to be in excellent agreement with predictions based on a theory for Fresnel reflection coefficients of an interface with subwavelength inclusions. The theory can also be used to derive analytical expressions for the Fano parameters.