Heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the mushrooms and ¼ teaspoon each salt and pepper and cook, tossing frequently, until golden and tender, 10 to 12 minutes. With a slotted spoon, transfer to a plate.
Remove the skillet from heat. Add the vinegar and scrape up any browned bits. Whisk in the mustard, sugar, and ¼ teaspoon each salt and pepper; gradually whisk in 2 tablespoons of the remaining oil. Add the frisée, radicchio, and mushrooms and toss to coat.
Meanwhile, season the fish with ¼ teaspoon each salt and pepper. Heat the remaining tablespoon of oil in a large nonstick skillet over medium-high heat. Cook the fish until opaque throughout, 3 to 4 minutes per side. Serve with the salad.
