Making the miso black cod: Combine the sugar with 3 tablespoons water in a small saucepan and bring to a boil. Boil until the sugar dissolves and most of the water has evaporated, 2 to 3 minutes. Pour the resulting syrup into a large bowl. Add the mirin, sake and both misos, and whisk to blend. Let the miso marinade cool completely.
Add the marinade to a large self-sealing plastic bag. Add the black cod fillets to the bag and seal, removing as much air as possible. Marinate the cod in the fridge for 24 hours.
Preheat the oven to 400 degrees F.
Place the fillets on a nonstick baking tray or a baking tray lined with parchment paper. Spoon the marinade generously over each fillet and discard the remainder. Cook the cod in the center of the oven until firm and beginning to flake, about 8 minutes. Raise the oven to broil and cook until the fish has caramelized, about 3 minutes.
Making the dirty shiitake rice: Add the red and wild rice to a medium saucepan filled with salted boiling water. Turn the heat down to maintain a low boil, and cook until the wild rice is tender, about 30 minutes. Drain and set aside. (This cooking method prevents the wild rice grains from splitting, the way they do when steamed, and leaves them pleasantly al dente.)
Saute the garlic in the olive oil over medium-high heat until golden, about 30 seconds. Add the mushrooms and shallots and cook until the liquid has evaporated and the mushrooms have begun to brown, 5 to 7 minutes. Deglaze with the sake and scrape into a medium saucepan. Add the long grain rice and then the vegetable stock, ginger and some salt. Bring to a boil and immediately adjust the heat to low. Cover and cook for 15 minutes. Remove from the heat and let sit, covered, for another 5 minutes. Fluff the rice with a fork and fold in the shiitake powder. Season with salt and pepper.
Serve the miso black cod accompanied by the dirty shiitake rice.