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Heat a saucepan over medium-high heat and add 1 tablespoon of the olive oil, the garlic, and the red pepper flakes. Stir for 30 seconds. Add the tomatoes and 1/2 teaspoon of the salt. Simmer for 30 minutes.
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Meanwhile, working on a large surface covered in baking parchment or plastic wrap, lay down the chicken breasts. Using a sharp slicing knife, cut each piece in half horizontally through the middle.
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In a large prep pan or dish, combine the bread crumbs and 1/2 cup of the Parmesan cheese. Spread out to cover the whole bottom of the pan. Lay down as many chicken breasts as will fit on the mixture. Sprinkle salt and pepper over each piece and turn over, completely coating with the bread-crumb mixture. Repeat the process with the remaining pieces of chicken.
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Heat a large skillet over medium-high heat. Swirl in 2 tablespoons of olive oil, along with butter, to coat the pan. Add the chicken breasts in one layer and cook until golden, about 3 minutes on each side. Remove and repeat the process for the remaining chicken, adding a little oil and butter to the pan as needed.
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Preheat the oven to 400 degrees Fahrenheit. Spoon some tomato sauce into a greased 9 x 13-inch baking dish to cover. Layer in the chicken pieces and top with mozzarella slices. Spoon over about 1 1/4 cups more sauce, and sprinkle on the remaining 1/4 cup of Parmesan cheese. Bake until golden and bubbling, 30 to 35 minutes. Let rest for at least 15 minutes before serving.
