We report a theoretical study of two-dimensional periodic arrays of composite particles, consisting of two metallic nanodisks separated by a dielectric spacer, using an efficient and accurate multiple-scattering method, which was originally developed for systems of spherical particles and recently extended to non-spherical shapes. In particular, we discuss the plasmonic excitations in such isolated nanosandwiches and study the influence of geometrical parameters like the thickness of the dielectric spacer. Moreover, we investigate the interaction between such composite particles as they approach each other in a two-dimensional periodic lattice.