"As lasagnas go, this takes just about as much time as any. It is a modified version of my grandmother's recipe, so that it includes more vegetables, including eggplant. You can add whatever vegetables you want to the meat sauce. I grabbed what was seasonal and on sale, but I imagine this would be good with some spinach, broccoli, or diced tomato. Also, consider some crushed red pepper if you like a spicy sauce. Not a fan of that myself, but it would add a nice heat if that is your thing. I don't like salty sauce, so you may find you want more salt if you do prefer it that way. My oven runs hot. If necessary, please adjust your baking time to make sure the lasagna is heated all the way through."
Heat oil in a large pot over medium heat. Add onion and garlic; cook and stir until onions start turning clear, 5 to 10 minutes. Add ground beef; season with salt and pepper. Cook and stir until browned and crumbly, 5 to 7 minutes. Add crushed tomatoes. Fill the empty tomato can with water and pour into the sauce.
Stir eggplant, zucchini, mushrooms, carrots, tomato puree, 2 tablespoons salt, basil, black pepper, oregano, and bay leaf into the sauce. Bring to a boil; reduce heat to medium-low. Cook, stirring occasionally, until flavors are well blended, about 1 hour. Remove and discard bay leaf.
Bring a large pot of lightly salted water to a boil. Cook lasagna noodles in the boiling water, stirring occasionally until tender yet firm to the bite, about 8 minutes. Drain and lay the noodles out on paper towels to dry; remove and discard any broken pieces.
Combine ricotta and egg in a bowl; season with salt and pepper and whisk thoroughly.
Preheat the oven to 375 degrees F (190 degrees C).
Coat the bottom of a large baking pan with the sauce. Layer noodles, ricotta mixture, sauce, mozzarella cheese, and a sprinkle of Parmesan cheese on top. Repeat layering until all used up. Cover the pan with aluminum foil.
Bake in the preheated oven until bubbling furiously, 30 to 35 minutes. Uncover pan and increase oven temperature to 400 degrees F (200 degrees C). Continue baking until top is golden brown, 10 to 15 minutes more. Let lasagna sit for 10 to 15 minutes before serving.