Cut bananas in half crosswise, then lengthwise. Melt butter in a large skillet over medium heat. Add brown sugar and cook, stirring constantly, 2 minutes. Add bananas to skillet and sprinkle with cinnamon.
Remove skillet from heat. Stir in rum, and carefully ignite the flames just above the mixture with a long match. Let flames die down. Return to heat and cook 3 to 4 minutes or until soft. Serve with a scoop of ice cream.