In a shallow dish, combine the sake, mirin, brown sugar, soy sauce and yellow miso. Add black cod, turning to coat well. Refrigerate and let marinate at least 2 hours, but no more than 6 hours.
Preheat a broiler to medium.
When ready to serve, broil the cod, 6 inches from the heat, until fish is opaque and lightly caramelized on top, about 5 to 7 minutes.
In a medium saucepot over medium heat, add the miso paste and slowly stir in the boiling water. Add the ginger, garlic and vinegar and bring to a boil. Strain broth from solids and return the broth to the heat. Add the crab, mushrooms and the scallions. Cook until the crab is heated through and mushrooms are tender, about 3 minutes. Serve with the broiled cod topped with smelt egg, if using.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.