Combine butter, brown sugar, and white sugar in a 2-quart saucepan. Cook over medium heat, stirring occasionally, until mixture comes to a boil and sugars dissolve, 6 to 8 minutes. Mix hot butter mixture into ice cream in a large bowl. Fill each mug with 3/4 cup boiling water, 1/4 cup ice cream mixture, and 2 tablespoons rum. Sprinkle nutmeg on top.