Melt butter, brown sugar, and cinnamon in a heavy skillet over low heat, stirring until the sugar fully dissolves and the mixture bubbles gently, 3 to 4 minutes
Stir in banana liqueur, then add bananas cut side down and cook 1 to 2 minutes per side, spooning the sauce over the top as they cook
Remove from heat, pour in the rum, let it warm 30 seconds, then tilt toward flame or use a long match to ignite. Flames will subside in 30 to 45 seconds
Plate bananas over ice cream and spoon the sauce generously over the top. Serve immediately