Stir sake, mirin, miso, and brown sugar together in a medium bowl until well mixed. Stir in chopped green onions and ginger.
Pat fish dry with paper towels, then lay in a glass baking dish just large enough to hold fish in a single layer. Pour in marinade and turn to coat. Chill at least 5 hours and up to 3 days, turning fillets over occasionally.
Preheat broiler with a rack set 6 to 7 in. from heat. Line a large rimmed baking sheet with foil. Lift half of fish from marinade, letting excess drip off, and add to pan skin (or skinned, flat) side down. Broil until well and evenly browned, 6 to 8 minutes (if only spots are browning after 6 minutes, lower heat to 500° and cook 1 to 2 minutes more). With a thin metal spatula, carefully loosen fillets from foil (the skin will most likely stay behind) and transfer to plates.
Serve fish sprinkled with sliced green onion.