1. Cook lasagne noodles in plenty of boiling salted water until al dente, or firm to the bite, about 12 minutes; drain. Let noodles sit in bowl of cool water until ready to use.
2. Steam the broccoli in a rack set over 1 inch simmering water, covered, until crisp-tender, about 5 minutes. Cool; coarsely chop; set aside.
3. Heat 2 tablespoons of the oil in a medium skillet; add onion; sauté until tender. Add 1 teaspoon of the garlic; sauté 1 minute.
4. In a bowl, combine the chopped broccoli, half the sautéed onion, the ricotta, eggs, 2 tablespoons of tile Parmesan cheese, 2 tablespoons of the parsley, and a pinch each of black pepper and nutmeg; set aside.
5. Add the chopped carrots to the onion remaining in the skillet; sauté, stirring, over medum heat until crisp-tender; stir in the tomatoes. Cook, stirring, until boiling gently. Boil, stirring, until sauce is thickened, about 10 minutes; stir in 1 tablespoon of the remaining parsley, the basil, and salt and pepper to taste. Transfer to a small bowl. Wipe out skillet.
6. Heat the remaining 2 tablespoons oil in the skillet; add the mushrooms. Sauté, stirring, over medium neat, until liquid has evaporated, about 5 minutes. Add green or red bell pepper; sauté until crisp-tender, about 3 minutes. Stir in the remaining 1 teaspoon garlic; sauté 1 minute. Stir in the remaining 1 tablespoon parsley and the lemon juice; season with salt and pepper.
7. Heat oven to 350°F. Select a shallow 4-quart baking dish approximately 10 × 14 inches. Spoon about 1/2 cup of the tomato sauce over the bottom of the dish. Lift the lasagne noodles from the water individually and blot dry on paper toweling. Arrange a single layer of noodles in dish.
8. Spread the mushroom layer over the noodles; sprinkle with 1 cup of the mozzarella and 1 tablespoon of the remaining Parmesan. Top with a second layer of noodles. Spread with the ricotta and broccoli mixture; sprinkle with ½ cup of the mozzarella. Top with a third layer of noodles. Spread with the tomato sauce. Sprinkle with the remaining 1 tablespoon Parmesan and the remaining 1 cup shredded mozzarella.
9. Bake until cheese is melted and bubbly, about 40 minutes. Let stand at least 15 minutes before serving.