This letter describes an extension of the classic Lazy Probabilistic Roadmaps algorithm (Lazy PRM), which results from pairing PRM and a novel Branch-And-Cut (BC) algorithm. Cuts are dynamically generated constraints that are imposed on minimum cost paths over the geometric graphs selected by PRM. Cuts eliminate paths that cannot be mapped into smooth plans that satisfy suitably defined geometric and differential constraints. We generate candidate smooth plans by fitting splines to vertices in a minimum-cost path. Plans are validated with a recently proposed algorithm that maps them into finite traces, without the need to choose a fixed discretization step. A trace records the exact sequence of constraint boundaries crossed by the plan, modulo arithmetic precision. We evaluate several planners using our methods over the recently proposed BARN benchmark, reporting evidence of the scalability of our approach.