Once the oil is hot, cook the chicken cutlets until golden brown and cooked through, 3-5 minutes per side depending on how thick they are. Cook them in two batches, transferring the cooked cutlets to a paper towel-lined plate while the others, adding more oil to the pan between batches if need be.
While the chicken is cooking, place a medium-size pot on the stove over medium heat. Add 2 tablespoons EVOO, 2 turns of the pan. Add garlic, crushed red pepper flakes and chopped onion. Cook 5-6 minutes, stirring often, until the veggies have softened. Add the tomatoes and heat them through, about 2 minutes. Remove the pot from the heat and stir in the basil, then season the sauce with salt and freshly ground black pepper.
Layer the chicken with the sauce in a casserole dish. Top the casserole with the remaining grated Parmigiano and mozzarella. Brown the chicken parm casserole under the broiler until the cheese has melted and is lightly golden brown, about 3 minutes. Garnish with some chopped basil before serving.