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 ½ minutes.
Flip bananas, and cook 2 ½ 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.
