Put 2 tablespoons oil in a large skillet over medium heat. When it’s hot, add the bread crumbs, sprinkle with salt and pepper, and cook, stirring often, until golden, 3 to 5 minutes. Add the red chile flakes and stir to combine, then transfer the bread- crumbs to a bowl and stir in the Parmesan. Taste and adjust the seasoning.
Put the remaining 2 tablespoons oil in the skillet and return to medium heat. When it’s hot, add the garlic and cook until fragrant but not browned, about 1 minute. Add the tomatoes and wine and a sprinkle of salt and pepper. Bring to a boil, then adjust the heat so the mixture bubbles steadily. Cook, stirring occasionally, until the tomatoes break down a bit while you prepare the chicken.
Put the chicken on a flat surface and put your hand on top of one. Carefully slice the breast in half parallel to the cutting board to make 2 thin cutlets; repeat with the other breast.
When the sauce has thickened, nestle the cutlets into the skillet, adjust the heat to a steady but gentle bubble, and cover. Cook, undisturbed, until the chicken is opaque around the edges, 3 to 5 minutes. Turn, top each with some cheese, and cover again. Cook until the chicken is no longer pink inside and the cheese has melted, another 3 to 5 minutes. Taste and adjust the seasoning of the sauce. Serve the chicken topped with sauce and sprinkled with bread crumbs.