Vehicle routing problems (VRPs) are a class of combinatorial problems with wide practical applications. While previous heuristic or learning-based works achieve decent solutions on small problem instances of up to 100 customers, their performance does not scale to large problems. This article presents a novel learning-augmented local search algorithm to solve large-scale VRP. The method iteratively improves the solution by identifying appropriate subproblems and delegating their improvement to a black box subsolver. At each step, we leverage spatial locality to consider only a linear number of subproblems, rather than exponential. We frame subproblem selection as a regression problem and train a Transformer on a generated training set of problem instances. We show that our method achieves state-of-the-art performance, with a speed-up of up to 15 times over strong baselines, on VRPs with sizes ranging from 500 to 3000.