Broiling the fillets hot and fast delivers tender, flaky fish and browns the smoky chile compound butter with zero effort. Use cultured butter here; typically higher in butterfat, it has a greater depth of flavor and browns more quickly.
Stir together butter, paprika, smoked salt, and piment d’Espelette in a small bowl until combined. Spoon butter mixture into middle of a square of parchment paper or wax paper. Roll into a 6-inch log; twist ends to seal. Refrigerate until firm, about 1 hour.
Preheat broiler to high with oven rack 6 inches from heat. Rub 1 tablespoon butter mixture on a rimmed baking sheet, and top with fillets, leaving gaps between fillets to allow for even cooking. Dot each sole fillet with 1 1/2 to 2 tablespoons butter mixture.
Broil until fish is opaque and flaky, 8 to 10 minutes, basting with remaining butter mixture a couple of times during broiling. Remove fish from oven, and let rest on baking sheet 4 minutes. Divide fillets among plates; lightly sprinkle servings with smoked salt.
Prepare the butter ahead of time; refrigerate up to 1 week or freeze up to 1 month.
Fresh, lemony English sparkling wine.