1. Sprinkle the tomatoes on both sides with salt and a few grindings of black pepper. Then dip the tomato slices in the flour, coating each side thoroughly and very gently shaking off any excess. In a 12-inch heavy skillet, preferably of the nonstick variety, melt the butter over moderate heat.
2. When the foam subsides, add the tomato slices and cook them for about 5 minutes, or until they are lightly browned. Sprinkle the tops with half the brown sugar, carefully turn the tomatoes over with a spatula and sprinkle with the rest of the brown sugar. Cook for 3 to 4 minutes, then transfer the slices to a heated serving platter.
3. Pour the cream into the pan, raise the heat to high and bring the cream to a boil, stirring constantly. Boil briskly for 2 to 3 minutes, or until the cream thickens. Taste for seasoning, then pour over the tomatoes. Sprinkle with the finely chopped parsley.
4. NOTE: Traditionally, this recipe is made with green tomatoes; however, they are not easily available. If you can find them, cook them somewhat more slowly and for a few minutes longer on each side.
