Cut the bananas in half across and then lengthwise.

Melt the butter in a large skillet over medium heat. Add the brown sugar and cinnamon and cook, stirring, until the sugar dissolves, about 2 minutes. Add the bananas and turning, cook on both sides until the bananas start to soften and brown, about 3 minutes. Add the banana liqueur and stir to blend into the caramel sauce. Carefully add the rum and shake the pan back and forth to warm the rum and flame the pan. (Or, off the heat, carefully ignite the pan with a match and return to the heat.) Shake the pan back and forth, basting the bananas, until the flame dies.

Divide the ice cream among 4 dessert plates. Gently lift the bananas from the pan and place 4 pieces on each scoop of ice cream. Spoon the sauce over the ice cream and bananas, and serve immediately.