Melt butter in a saucepan over medium-low heat until butter smells toasted and is golden brown, about 1 minute. Turn off heat.
Line a baking sheet with a piece of aluminum foil. Place trout onto foil; open trout so skin sides are down. Drizzle each trout with about 1/2 teaspoon melted butter. Generously season with salt and black pepper.
Move an oven rack to 5 or 6 inches below heat source and preheat oven's broiler on high heat.
Broil trout until opaque and barely firm to the touch, 2 or 3 minutes. Remove from oven.
Return pan of melted butter over high heat; stir in lemon juice and parsley. Bring butter sauce to a boil; whisk to combine. Serve trout on plates and drizzle with butter sauce.
