In a 2-quart saucepan over high heat, heat crushed tomatoes, garlic, basil, salt, and pepper. Over high heat, heat pan to boiling; reduce heat to low. Simmer uncovered, 10 minutes to blend flavors.
Preheat the oven to 375 degrees F (190 degrees C). Grease a 12x8-inch baking dish. In a medium bowl, combine ricotta cheese and egg until well-mixed. Spoon some tomato sauce on the bottom of the dish to coat. Place 3 noodles lengthwise across the pan. Top with 1/3 of tomato sauce. With a spatula, spread 1/3 of ricotta mixture. Scatter 1/3 of carrots and spinach on ricotta and 1/3 of mozzarella cheese. Repeat with noodles, tomato sauce, ricotta, vegetables, and mozzarella two more times. Sprinkle with Parmesan cheese.
Cover dish with foil; bake 30 minutes. Uncover dish; bake 10 minutes longer. Let stand 15 minutes before cutting.