Slice the chicken breasts in half horizontally to make 4 thin cutlets. Put between 2 pieces of plastic wrap and pound with a meat mallet or heavy skillet until 1/4 inch thick. Heat 1 tablespoon olive oil in a medium pot over medium heat. Add the garlic and red pepper flakes and cook, stirring, until the garlic softens, about 1 minute. Add the tomato puree, then rinse out the can with 1 cup water and add to the pot. Simmer until the sauce just thickens, about 10 minutes. Whisk in 1/4 cup parmesan and the basil. Spread 1/2 cup of the sauce in a 3-quart flameproof baking dish. Preheat the broiler. Meanwhile, combine the flour and 1/2 teaspoon each salt and pepper in a shallow dish. Beat the eggs with 1/4 cup parmesan in another shallow dish. Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Dredge 2 chicken cutlets in the flour, tapping off the excess, then dip in the egg mixture. Carefully lay in the skillet and fry until golden brown and just cooked through, 2 to 3 minutes per side. Transfer to the baking dish. Add the remaining 2 tablespoons olive oil to the skillet and repeat with the remaining 2 chicken cutlets. Add to the baking dish. Combine the mozzarella and the remaining 2 tablespoons parmesan in a bowl. Sprinkle half of the cheese mixture on the chicken, then top with the remaining sauce and the remaining cheese mixture. Broil until the cheese melts, about 2 minutes.