Preheat oven to 425°F. Using aluminum foil, build a 10-inch square "pan" with a 1-inch-tall edge on all sides. Place foil square on 1 side of a rimmed baking sheet. Place tomatoes, garlic, 2 teaspoons oil, and 1/4 teaspoon each of the salt and pepper in foil square; stir to combine. Bake in preheated oven 20 minutes.
While tomatoes bake, cook linguine according to package directions. Drain; cover to keep warm.
Cut each chicken breast in half horizontally, parallel to work surface, to form 4 thin cutlets. Place eggs in a shallow dish. Place panko in a second shallow dish. Dip chicken cutlets, 1 at a time, in egg, letting excess drip off. Dredge in panko, pressing firmly to adhere to both sides of cutlets. Place coated cutlets on baking sheet beside foil square. Bake chicken and tomatoes until chicken is done and tomatoes are lightly browned, about 15 minutes.
Place tomato mixture, tomato paste, and remaining 2 tablespoons oil in a food processor; pulse until finely chopped and almost smooth, 3 to 4 times. Set aside.
Increase oven temperature to broil with oven rack 6 inches from heat. Top each cutlet evenly with cheeses. Broil until cheese is bubbly, about 3 minutes.
Serve chicken over pasta, and drizzle evenly with tomato sauce. Sprinkle servings with basil.