Bananas are cooked in a buttery brown-sugar sauce; rum is added and ignited just before serving (the flames burn off the alcohol, leaving the flavor behind). Ice cream melts into the warm glaze.
Melt butter with sugar in a 12-inch skillet over medium-high heat. Cook, swirling skillet occasionally, until color deepens, about 3 minutes. Add bananas, cut sides down. Cook, swirling occasionally to make sure bananas are coated, 3 1/2 minutes. Flip bananas, and cook 2 1/2 minutes more.
Remove skillet from heat. Add rum. Return to medium heat, and cook about 10 seconds to allow rum to heat up. If using a gas stove, carefully tip the pan away from you until the vapors from the rum ignite. (Alternatively, light the rum with a long match.) When the flames subside, remove pan from heat. Serve bananas topped with vanilla ice cream.