In a colander set in the sink, sprinkle 2 teaspoons salt on the eggplant and let drain for 15 minutes. Rinse the eggplant and squeeze dry with your hands.
In a large skillet, heat the oil over medium-high heat until rippling. Add the eggplant and cook, stirring occasionally, until softened and lightly browned, about 20 minutes.
Meanwhile, pat dry the flounder fillets and season both sides with the paprika and 1/2 teaspoon each salt and pepper.
In a large nonstick skillet, melt the butter over medium-high heat. Add the flounder in a single layer, cook for 4 minutes, then flip carefully and continue cooking until lightly golden, about 4 minutes more. Divide among 4 dinner plates.
Stir the tomatoes into the eggplant and cook until just heated through. Season with the thyme and 1/2 teaspoon pepper. Spoon over the flounder.