Preheat broiler. Keeping skin intact, mash avocado flesh. Broil mashed avocados in skin shells for 2 minutes. Set aside on a warm plate.
Saute fish pieces in clarified butter on high heat. Season with white pepper. Stir in brown sugar. Add shrimp, pimento, and garlic. Pour in rum, shaking pan constantly. Add dill. Add butter and remove pan from heat, letting butter melt into the sauce.
Spoon fish mixture onto avocado halves, sprinkle with grated cheese and broil again for a few minutes, until cheese begins to brown.