We study a model of a composite system constructed from a ”pairing layer” of disconnected attractive-U Hubbard sites that is coupled by single-particle tunneling, t_⊥ , to a metallic layer.  T_c reaches a maximum for intermediate values of t_⊥ , which we calculate using a combination of mean-field, classical and quantum Monte Carlo methods. The maximal T_c scales as a fraction of the zero temperature gap of the attractive sites when U is smaller than the metallic bandwidth, and is bounded by the maximal T_c of the two-dimensional attractive Hubbard model for large U . An effective weak coupling theory is constructed in order to explain the deviations of the Monte Carlo simulations from the mean-field calculation.