Preheat the oven to 375°F and lightly spray a 9 x 13 x 2.5-inch baking dish with nonstick spray; set aside.
Heat a large skillet over medium-high heat and add 1 tablespoon olive oil.
Add the diced zucchini, chopped mushrooms, diced carrots, and diced onion to the skillet and cook 4–5 minutes, stirring occasionally, until the vegetables are tender.
Add the spinach and cook about 1 minute more, until wilted.
Stir in the crushed tomatoes, tomato sauce, tomato paste, Italian seasoning, granulated sugar, and chopped basil; reduce heat to low and simmer about 10 minutes. Season with salt and ground black pepper to taste.
Meanwhile, bring a large pot of water to a rolling boil. Cook the lasagna noodles until al dente (about 8 minutes), then drain and rinse with cold water to stop cooking.
In a medium bowl, combine the ricotta cheese, grated Parmesan, and the large egg until smooth.
Assemble the lasagna: spread 1 cup of the vegetable-tomato mixture on the bottom of the prepared baking dish. Lay a single layer of noodles over the sauce, spread 1/2 cup of the ricotta mixture over the noodles, and sprinkle a handful of shredded mozzarella. Repeat layers until ingredients are used, finishing with the remaining mozzarella on top.
Cover the dish with foil and bake for 20 minutes. Remove the foil and bake an additional 20–25 minutes until heated through and the top is bubbly and browned.
Let the lasagna rest for 10 minutes before slicing and serving.