Heat the oven to 450°F
In a large roasting pan, combine bell peppers, eggplant, zucchini, squash, mushrooms, shallots, and rosemary. Toss with the oil, 1 ¼ tsp salt, ¼ tsp pepper. Roast, stirring occasionally, until softened and beginning to brown, about 25 minutes.
Discard the rosemary sprig, if using. When the shallots are cool enough to handle, peel and chop them. Combine the chopped shallots, ricotta, 1 cup of mozzarella, 2 tablespoons of the parmesan, nutmeg, egg, ½ tsp salt, 1/8 tsp pepper.
Cook the lasagna noodles, slightly underdone. Drain with cold water and dry on paper towels.
Oil a 9x13” baking dish. Spread ½ cup of tomato sauce over the bottom of dish. Lay several sheets of pasta, slightly overlapping, on the sauce. Dot with half the ricotta mixture. Spread half of the roasted vegetables on top of the cheese. Pour about 1 cup of sauce over the vegetables. Top the sauce with a second layer of pasta. Dot with the remaining ricotta mixture and cover with the remaining vegetables and about 1 cup of sauce. Top the sauce with a final layer of pasta and the remaining sauce. Sprinkle with the remaining 1 cup of mozzarella and 2 tablespoons of parmesan.
Bake at 350F until golden and bubbling, about 45 minutes. Let rest 15 minutes before cutting into squares.