Wash, peel and cut the broccoli into florets. Peel and slice the mushrooms. Wash the eggplant and zucchini and cut into cubes. Cut into fresh tomatoes, place in boiling water, remove, peel and cut into cubes. Peel and finely chop the garlic.
Heat the olive oil in a large pan. Fry the garlic briefly, then add the mushrooms, eggplant, zucchini and broccoli one after the other. Add a little water and cook with the lid closed. Then add tomatoes (fresh and canned) and cook until the fresh tomato has a soupy consistency (it can take a long time, it improves the tomato taste). Season the vegetables with salt, pepper and sugar.
While the vegetables are cooking, grate the parmesan and mozzarella. Leave the butter in a saucepan over medium heat, while stirring with a whisk, add just enough flour so that it remains thick but runny. Add the milk to the butter and flour mixture while stirring constantly, bring to the boil, season with salt, pepper and nutmeg.
Grease a lasagne dish with butter. Spread 1/3 of the lasagne plates on the floor, on top of them 1/2 of the cooked vegetables. Put 1/3 of the white sauce on top. Then spread 1/3 of the lasagna plates again, then the remaining vegetables, then 1/3 of the white sauce, then the remaining lasagne plates, then the remaining white sauce. Finally spread the cheese over the lasagna.
Put the top / bottom heat in the preheated oven at approx. 180 ° C for about 45 minutes on the middle rack of the oven, until the cheese is nice and brown and the lasagne plates are cooked through.