In a large heavy skillet, melt the butter over medium heat. In 2 batches, add the plantains and fry them in a single layer for about 4 minutes on each side. Remove with a slotted spoon and keep warm in a bowl while you fry the second batch.
Transfer the second batch of plantains to keep warm with the first. In the same pan, combine the brown sugar, butter, and cream. Stir over medium heat until the sugar has dissolved, then increase the heat to high and bring the mixture to a vigorous boil. Cook for 2 minutes, until slightly thickened. Return the plantains to the pan and stir to coat them evenly with the sauce. Pour the rum over the top and immediately light it with a match. Quickly spoon the flaming plantains over bowls of vanilla ice cream and serve.