Melt the butter in a large skillet over medium-low heat. Add the brown sugar and stir until the sugar completely dissolves, about 2 minutes. Lay the bananas in the pan and cook on both sides until they become slightly soft and begin to brown, about 3 minutes. Remove the pan from the heat and add the banana liqueur and then the rum and cinnamon. Tip the pan slightly and carefully ignite the alcohol with a long kitchen match or kitchen lighter to flambe. Put the pan back on the heat and shake it back and forth basting the bananas, until the flame dies out.
Divide the ice cream among dessert bowls. Gently lift the bananas from the pan and place 2 pieces on the ice cream. Spoon the sauce over the ice cream and serve immediately.