Bring a pan of salted water to the boil, and cook broccoli until tender with a bite, about 10 minutes. Drain and chop.
Bring another large pan of water to the boil, and cook the lasagne sheets according to the packet instructions. Drain.
Meanwhile, wash and cut the carrot, aubergine, potatoes, red peppers and courgette into Julienne style strips.
Heat 1 tablespoon of the oil in a frying pan. Cook and stir the garlic just until fragrant, about 1 minute. Stir in mushrooms, courgettes and salt. Set aside.
Heat the remaining 2 tablespoons oil in another frying pan. Add onions, leeks and celery and cook and stir for 10 minutes. Stir in the mushroom mixture, chopped broccoli, and all of the remaining vegetables and cook until heated through, about 5 minutes. Season with salt and pepper.
Stir the bechamel sauce into the vegetables.
Grease a glass baking dish with oil, add 1/3 of the vegetables and sprinkle with Parmesan cheese. Add a layer of lasagne sheets.
Continue layering until you run out of ingredients, ending with bechamel and Parmesan.
Bake in the preheated oven for 1 hour. Let it rest for at least 10 minutes before slicing and serving.