Preheat the oven to 180 degrees C. Lightly coat a 20cm square baking dish with cooking spray.
In a large bowl, toss together the zucchini, mushrooms, spinach, garlic, red onion, 1/2 cup mozzarella cheese and feta cheese. Drizzle with olive oil and balsamic vinegar, and stir in basil, oregano, sugar, salt and pepper. Stir the mixture until evenly blended.
Place a layer of lasagne into the bottom of the prepared pan. Make a layer of tomato slices over the lasagne. Spread a generous amount of the spinach mixture over the tomatoes. Don't worry, it shrinks a lot while cooking. Lay slices of tomatoes over the spinach mixture, then another layer of lasagne. Start with another layer of tomatoes on top of the noodles, and repeat layering until the dish is heaped with lasagne, ending with the vegetable mixture. Sprinkle remaining cheese on the top.
Bake for 35 to 45 minutes in the preheated oven, until lasagne and vegetables are tender. Let stand for a few minutes to set, then slice and serve.