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Achieving a proper balance between planning quality, safety and efficiency is a major challenge for autonomous driving. Optimisation-based motion planners are capable of producing safe, smooth and comfortable plans, but often at the cost of runtime efficiency. On the other hand, naively deploying trajectories produced by efficient-to-run deep imitation learning approaches might risk compromising safety. In this paper, we present PILOT -- a planning framework that comprises an imitation neural network followed by an efficient optimiser that actively rectifies the network's plan, guaranteeing fulfilment of safety and comfort requirements. The objective of the efficient optimiser is the same as the objective of an expensive-to-run optimisation-based planning system that the neural network is trained offline to imitate. This efficient optimiser provides a key layer of online protection from learning failures or deficiency in out-of-distribution situations that might compromise safety or comfort. Using a state-of-the-art, runtime-intensive optimisation-based method as the expert, we demonstrate in simulated autonomous driving experiments in CARLA that PILOT achieves a seven-fold reduction in runtime when compared to the expert it imitates without sacrificing planning quality.