The Reissner-Mindlin plate models thin plates. The condition numbers of finite element approximations of these plate models increase very rapidly as the thickness of the plate goes to 0. A Balancing Domain Decomposition by Constraints (BDDC) De Luxe method is developed for these plate problems discretized by Falk-Tu finite elements. In this new algorithm, subdomain Schur complements restricted to individual edges are used to define the average operator for the BDDC De Luxe method. It is established that the condition number of this preconditioned iterative method is bounded by C(1 + log (H/h))^2 if t, the thickness of the plate, is on the order of the element size h or smaller; H is the maximum diameter of the subdomains. The constant C is independent of the thickness t as well as H and h. Numerical results, which verify the theory, and a comparison with a traditional BDDC method are also provided.