Whisk miso, 1 cup sugar, sake, and mirin together in a saucepan; bring to a boil. Reduce heat to low and simmer, stirring occasionally, until marinade has reduced, about 45 minutes. Transfer marinade to a shallow dish and cool completely, 30 minutes to 1 hour. Place butterfish in the marinade and refrigerate, 8 hours to overnight.
Combine shallot, bay leaf, peppercorns, vinegar, and wine in a saucepan; bring to a boil. Reduce heat to medium and simmer until only about 1 tablespoon liquid remain, 2 to 3 minutes. Pour cream into mixture and simmer until cream mixture has reduced by half, 2 to 3 minutes. Increase heat to medium-high and rapidly whisk butter pieces, 1 at a time, until beurre blanc is smooth and thickened. Strain beurre blanc through a mesh strainer to remove spices.
Stir lemon juice into beurre blanc until lemon sauce is evenly mixed; fold in parsley.
Combine soy sauce and 1 cup sugar in a saucepan over low heat; cook and stir until sugar is dissolved and sweet soy sauce has reduced to 1 cup, 15 to 20 minutes.
Heat a skillet over medium heat for 2 to 3 minutes. Remove butterfish from marinade, discarding unused marinade. Cook butterfish in the hot skillet until fish flakes easily with a fork, 3 to 4 minutes per side. Transfer butterfish to a warmed plate and drizzle lemon sauce and sweet soy sauce around fish.