On a cutting board, place the chicken breasts between a piece of plastic wrap and pound the chicken out until it is 1/2-inch thick.
In 3 separate shallow dishes, place the flour, eggs, and bread crumbs. Season the chicken with salt and pepper. Dredge the cutlets through the flour, then eggs, and then the bread crumbs.
In a large skillet over medium-high heat, heat the vegetable oil. Place the cutlets in the oil and fry until brown on each side, about 2 to 3 minutes per side. Place the cooked cutlets in a baking dish. Spoon tomato sauce over each cutlet and sprinkle evenly with the cheeses. Bake for 10 to 15 minutes, until the cheese is melted and bubbly.
In a large saucepan over medium-high heat, add the olive oil and saute the onion and garlic. Add salt, pepper, red pepper flakes and cook until fragrant and tender; about 3 minutes. Add tomatoes, basil, and let simmer for 30 to 45 minutes. Season, to taste, with salt and pepper.