Rinse trout; pat dry. Place trout in a shallow dish. Add milk. Sprinkle with salt and pepper. Turn the fish once.
Place flour in a shallow dish. Remove fish from milk. Dip in flour to coat all sides.
In a 12-inch skillet, heat oil over medium-high heat. Add fish and cook for 5 minutes. Turn and cook for 5 to 8 minutes more or until fish flakes easily with a fork. Transfer to a hot platter; keep warm. Scrape and wipe out skillet.
Melt 1 tablespoon of the butter in the skillet. Cook mushroom slices in butter, stirring often, until all liquid evaporates and mushrooms are brown.
Melt the remaining butter in skillet. Serve mushrooms and butter over fish. Serve with lime wedges. Makes 4 servings.