1. Rinse green tomatoes; cut crosswise into 1/2-inch-thick slices, discarding tops and bottoms. Lightly sprinkle both sides of slices with salt and pepper. Lay in a single layer on paper towels. Let stand 5 to 10 minutes.
2. Meanwhile, cut red tomatoes in half and squeeze juice and flesh into a fine strainer set over a bowl; press enough tomato solids through strainer to make about 1/3 cup liquid.
3. Coat green tomato slices in semolina flour or yellow cornmeal. Pour 2 tablespoons olive oil into a 10- to 12-inch nonstick frying pan over medium heat. Cook tomato slices in a single layer until golden brown on both sides, 8 to 10 minutes total. Stack slices on two dinner plates.
4. Add the 1/3 cup tomato juice and solids, vinegar, and butter to pan; stir constantly over medium-high heat until butter is melted and incorporated into juices, 1 to 2 minutes. Add parsley and salt and pepper to taste. Spoon sauce over tomato stacks.
5. Mix frisée with 1 teaspoon olive oil; mound equally on tomato stacks.