Adjust oven rack to upper-middle position and heat oven to 475 F. Heat 2 tablespoons olive oil in medium saucepan over medium heat until shimmering. Add onion and cook until softened, about 3 minutes. Stir in garlic and cook until fragrant, about 30 minutes. Add tomatoes and simmer until slightly thickened, about 5 minutes. Off heat, stir in half of basil. Transfer half of sauce to 8-inch square baking dish.
Combine mozzarella and Parmesan in medium bowl; reserve 1/2 cup. Add remaining basil to remaining cheese mixture. Pat cutlets dry with paper towels and season with salt and pepper. Top cutlets with basil-cheese mixture, leaving 1-inch border at bottom of cutlets. Roll up tightly and arrange, seam side down, in prepared baking dish.
Toss bread crumbs with remaining oil in small bowl. Top chicken with reserved cheese, remaining sauce and bread crumb mixture. Bake until chicken is cooked through and crumbs are golden, 15 to 20 minutes. Serve.
Test Kitchen Note: Cook's Country's favorite brand of diced tomatoes is Hunt's.