Put the fish in a shallow dish and add the milk. Turn the fish to coat on both sides. Set aside.
Peel the lime. Cut the flesh from the white pulpy sections to make wedges. Set aside.
Remove the fish from the milk without patting dry. Dip the fish in the flour to coat on all sides. Shake off any excess flour.
Heat the oil in a skillet large enough to hold the fish in one layer. Add the fifh and cook over moderately high heat 2-3 min. or until golden brown on one side Turn and cook on the other side, basting often, about 8-10 min.
Transfer the fish to a warm serving platter and keep hot.
Heat 1 tbsp of butter in another skillet and add the mushrooms. Cook, shaking the skillet and stirring, until the mushrooms are soft. Continue cooking until the liquid evaporates and the mushrooms are browned, about 3 min.
Arrange the mushrooms neatly over the trout. Arrange the lime wedges symmetrically between the mushroom slices. Sprinkle any accumulated lime juice over the trout.
Heat the remaing butter in the skillet in which you cooked the mushrooms. Continue cooking untilo the butter is lightly brown. Pour this over the trout. Sprinkle with parsley.