Bring the sake and mirin to a boil in a medium saucepan over high heat. Boil for 20 seconds to evaporate the alcohol. Turn the heat down to low, add the miso paste, and whisk. When the miso has dissolved completely, turn the heat up to high again and add the sugar, whisking constantly to ensure that the sugar doesn't burn on the bottom of the pan. Remove from heat once the sugar is fully dissolved. Cool to room temperature.
Pat the black cod fillets thoroughly dry with paper towels. Slather the fish with the miso marinade and place in a non-reactive dish or bowl and cover tightly with plastic wrap. Leave to marinate in the refrigerator for at least 3 hours and up to 3 days.
Preheat oven to 425°F. Heat an oven-proof skillet over high heat on the stovetop. Lightly wipe off any excess miso clinging to the fillets, but don't rinse it off. Add a little oil to the pan, then place the fish skin-side-up on the pan and cook until the bottom of the fish browns and blackens in spots, about 3 minutes. Flip and continue cooking until the other side is browned, 2 to 3 minutes more. Transfer to the oven and bake for 5 to 10 minutes, until fish is opaque and flakes easily.
Serve over a bed of white rice and charred broccolini