In a small bowl, whisk together brown sugar, soy sauce, hot water, and miso paste. Whisk until combined.
Lightly oil a baking dish and place salmon in it, cover with miso mixture.
Broil in oven for 10 minutes. During the 10 minutes, baste the fish with the miso sauce twice.
While fish is broiling, prepare the sauce by melting 1 tbsp of butter over medium heat in a small sauce pan. Add in the ginger and garlic paste. Cook for 2 minutes until fragrant and combined.
Add Mirin and bring to a boil for 3 minutes.
Add in heavy cream and return to boil for 2 minutes.
Cook until you have reduced the mixture by half, which should take 2-4 minutes.
Once reduced, whisk in cold butter, one piece at a time, and once all is combined and sauce has thickened, remove from heat.
Whisk in lime juice. Taste, and season with salt if necessary.
Serve by topping rice with the sauce, and placing salmon on top.
Add steamed snow peas to the side.