Preheat the oven to 180° C. Boil the cauliflower and broccoli for 5 minutes, then transfer them to iced water. Drain well and lightly crush them with a fork.
Heat 2 tbsp oil in a large pan and add the onion. After about a minute, add the besciamella sauce and 3/4 of the Parmesan, making it melt.
Lay 4 of the lasagne sheets in a generously oiled baking dish.
Spread the vegetables over the lasagne sheets, then add some of the goat cheese in crumbles, then cover with a layer of besciamella. Top with 4 sheets of pasta, repeat and finish with a layer of pasta.
Cover the top of the pasta with the remaining Parmesan and bake for about 15 minutes, until bubbling and golden.
Serve hot or warm.