FOR THE FISH: Adjust oven rack to lower-middle position, set 4 heatproof dinner plates on rack, and heat oven to 200 degrees. Place flour in large baking dish. Season both sides of each fillet generously with salt and pepper; let stand until fillets are glistening with moisture, about 5 minutes. Coat both sides of fillets with flour, shake off excess, and place in single layer on baking sheet. Heat 1 tablespoon oil in 12-inch nonstick skillet over high heat until shimmering, then add 1 tablespoon butter and swirl to coat pan bottom; when foaming subsides, carefully place 2 fillets in skillet, bone-side down. Immediately reduce heat to medium-high and cook, without moving fish, until edges of fillets are opaque and bottom is golden brown, about 3 minutes. Using 2 spatulas, gently flip fillets (see illustration below) and cook on second side until thickest part of fillet easily separates into flakes when toothpick is inserted, about 2 minutes longer. Transfer fillets, one to each heated dinner plate, keeping bone- side up, and return plates to oven. Wipe out skillet and repeat with remaining 1 tablespoon each oil and butter and remaining fish fillets.
FOR THE BROWNED BUTTER: Heat butter in 10-inch skillet over medium-high heat until butter melts, 1 to 1 1/2 minutes. Continue to cook, swirling pan constantly, until butter is golden brown and has nutty aroma, 1 to 1 1/2 minutes; remove skillet from heat. Remove plates from oven and sprinkle fillets with parsley. Add lemon juice to browned butter and season to taste with salt; spoon sauce over fish and serve immediately with lemon wedges.