Red miso gives the salmon a certain oomph. It is less salty than yellow miso, and goes best with fish. Be sure to use high-quality sake.
Combine the brown sugar, soy sauce, hot water, and miso and whisk until the sugar has dissolved. Set aside.
Choose a gratin dish large enough to hold the salmon without crowding, and lightly coat it with oil. Arrange the salmon fillets in the dish, and spread the miso mixture evenly over each fillet. Broil the fillets, basting with the miso mixture once or twice, for 7 to 8 minutes for medium rare, or 10 minutes for medium, or until the fillets flake easily when prodded with a fork.
Meanwhile, in a small saucepan over medium heat, soften the ginger and shallots in the butter for 3 to 4 minutes. Pour in ½ cup of the sake, bring the mixture to a boil, and reduce to about 3 tablespoons, about 3 minutes. Pour in the half-and half, return to a boil, and reduce by about half, about 2 minutes. Increase the heat to medium-high, and, whisking constantly, add the butter, one cube at a time. The butter should create a thick and creamy sauce. When all of the butter has been incorporated, remove the saucepan from the heat and whisk in the remaining 1 teaspoon of sake and the lime juice. Season to taste with salt and white pepper. Using an immersion blender or in a stand blender purée the sauce, in batches, until very smooth.
Divide the sake butter between two plates, top with the basmati rice and then the salmon, and serve at once.