We report theoretical and experimental investigations of the optical response of two-dimensional periodic arrays of rectangular gold nanopatches grown on a monolayer graphene placed on a glass substrate. We discuss the numerical analysis and optical characterization by means of reflection spectra and show that rectangular nanopatches display a polarization-dependent response, at normal incidence, which leads to double plasmonic resonances due to the Wood anomaly. We detail the fabrication process highlighting how the resist primer and the adhesion layer can reduce and impede the graphene doping due to the environment and to the nanopatches, respectively, by means of Raman spectroscopy.