Preheat the oven to 425 degrees F. Season the chicken on both sides with some salt and pepper. Spread the flour in a shallow bowl or plate. Season with some salt and pepper.
Heat a large skillet over medium-high heat. Add 2 tablespoons of the olive oil and heat. Working with 2 or 3 pieces at a time so as not to crowd the pan, dredge both sides of the chicken cutlets in the seasoned flour and tap off the excess. Cook the chicken until golden brown on both sides, 1 to 2 minutes per side. Remove to a plate and repeat with the remaining cutlets.
Reduce the heat to medium and add the remaining 2 tablespoons olive oil to the skillet. Add the garlic and thyme and cook until the garlic is just golden on the edges, about 1 minute. Add the tomatoes, rinse the can with 1/2 cup water and add that to the skillet. Add the sugar, red pepper flakes and 1/2 teaspoon salt. Bring the sauce to a rapid simmer and cook, stirring occasionally, until slightly thickened, about 10 minutes.
Tear the basil into rough shreds with your hands and stir into the sauce. Brush a 9-by-13- inch baking dish with some olive oil and ladle half of the sauce into the dish. Nestle the chicken into the sauce and top with the remaining sauce. Lay the sliced mozzarella over the chicken and sprinkle the Parmesan over all. Bake until the cheese is browned and bubbly, 15 to 20 minutes. Serve hot.