Bring a large pot of water to a boil. Add the ravioli and cook according to the package directions. Drain and rinse under cold water until cool enough to handle.
Position an oven rack in the middle of the oven and preheat to 400 degrees F. Brush the cups of a 12-cup muffin tin with 2 tablespoons of the oil. Set aside.
Combine the tomatoes, Italian seasoning, carrot, garlic and remaining 2 tablespoons oil in a blender. Blend until well combined but not entirely smooth, about 20 seconds. Season with salt and pepper.
Spoon 1/2 teaspoon sauce into each oiled cup, top with a ravioli, then with 1 teaspoon sauce and 1 teaspoon mozzarella. Repeat twice so that each cup has 3 raviolis layered on top of each other with sauce and mozzarella. Evenly divide the remaining sauce among the cups. Top each with 1 tablespoon mozzarella and 1/4 teaspoon Parmesan. Bake until the top of each lasagna is golden brown and bubbly, 18 to 20 minutes.
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