In a medium saucepan, bring the mirin and sake to a boil. Remove from the heat, add the miso and stir until dissolved. Add the sugar, return to a low heat, and stir until dissolved. Remove from the heat and let cool completely.
Place the fish in a non-reactive baking dish. Cover with the cooled marinade, cover with plastic, and let marinate overnight refrigerated, or at least 4 hours.
Preheat the broiler.
Remove the fish from the marinade and place on a broil pan. Blot the miso from the fillets with a kitchen towel. Broil until lightly browned and the fish is opaque in the center, 6 to 10 minutes, depending upon the size of the fillets and the height of the broiler element.
Remove from the oven. Spoon rice onto each of 4 plates and top with the fillets. Arrange the cucumber relish to the side of the fish, and garnish each serving with the green onions and toasted sesame seeds.
Yield: 4 servings