Preheat the oven to 375 degrees F (190 degrees C). Grease an 8x12-inch baking dish.

Combine tomatoes, garlic, basil, salt, and black pepper in a medium saucepan over high heat; bring to a boil. Reduce heat to low; simmer to blend flavors, about 10 minutes. Combine ricotta cheese and egg in a bowl.

Lightly coat the baking dish with tomato sauce; place 3 noodles on top, lengthwise. Top with 1/4 remaining tomato sauce; spread 1/3 ricotta mixture on top with a rubber spatula. Scatter 1/3 each carrots and spinach on top, followed by 1/4 mozzarella. Repeat layers twice more. Place remaining 3 noodles on top; top with remaining 1/4 each tomato sauce and mozzarella. Sprinkle with Parmesan cheese.

Cover dish with aluminum foil; bake 30 minutes. Remove foil and bake until cheese is bubbling and lightly brown, about 10 minutes more. Let stand before cutting, 15 minutes.