Pat the fillets dry and cut a shallow cross into the skin. Thinly slice the ginger.
Put the water, sake, soy sauce, mirin, and sugar in a frying pan and bring to a boil over medium heat. Add the fish skin-side up with the ginger.
Cover with a drop lid and the pan lid, reduce to low, and simmer for 10 to 15 minutes depending on thickness.
Remove both lids and keep spooning the broth over the fish until it reduces to half or less and looks glossy. Watch closely at the end so the sugar does not burn.
Transfer the fish to plates with a wide spatula, spoon over the sauce, and add the simmered ginger (and shredded green onion if using).