Slice the bananas on a bias about 2 inches long.
Melt butter in a large saute pan over medium heat. Add the brown sugar and cinnamon and cook, stirring until the sugar dissolves, about 2 minutes. Add the bananas. Turn and cook on both sides. Cook until bananas are softened and slightly brown, about 2 minutes. Add the banana liqueur and stir to blend into the caramel sauce. Add the rum and shake the pan back and forth to warm the rum. Being very careful, flame the rum with a long tubed lighter. Shake the pan back and forth, basting the bananas, until the flame dies.
Divide the ice cream between 4 dessert bowls or plates Slightly flatten the tops. Gently lift the bananas from the pan and place over the ice cream. Spoon the sauce over all.