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.
Cut any hard crusts off the bread with a serrated knife, then cut the bread into cubes. Add the bread cubes to a food processor with the motor running, a few pieces at a time, and process to make coarse crumbs (you will have about 1 1/4 cups crumbs).
Heat 1 tablespoon olive oil in a large nonstick ovenproof skillet over medium-high heat. Add the breadcrumbs and cook, stirring, until golden and crisp, about 3 minutes. Add 2 tablespoons parmesan and cook, stirring, until melted, about 1 more minute. Transfer to a bowl and stir in 2 tablespoons basil; set aside. Preheat the broiler.
Return the skillet to medium-high heat and add the remaining 1 tablespoon olive oil. Sprinkle the chicken with 1/4 teaspoon salt, and pepper to taste; add to the skillet (in batches, if necessary) and cook until golden and just cooked through, 2 to 3 minutes per side. Transfer to a plate.
Add the tomato puree, then rinse out the can with 1 cup water and add to the skillet. Add the garlic, red pepper flakes and 1 tablespoon parmesan; reduce the heat to medium and simmer, stirring occasionally, until the sauce is slightly thickened, about 10 minutes. Stir in the remaining 1/4 cup basil and season with salt and pepper.
Return the chicken to the skillet, turning to coat with the sauce. Top with the remaining 1 tablespoon parmesan and all but a few tablespoons of the toasted breadcrumb mixture. Sprinkle with the mozzarella and the remaining breadcrumbs. Broil until the cheese melts, 1 to 2 minutes.