In a 12-inch skillet, melt the butter over medium heat. Stir in the brown sugar and cinnamon and cook until moist, about 1 minute. Add the banana liqueur and cook, stirring continuously, until the grainy texture disappears and the mixture rolls off the spoon in a steady stream, 1 to 2 minutes. Put the banana quarters rounded side down in the pan. Cook until the bananas begin to soften, 2 to 3 minutes. Turn the bananas onto their flat sides. Remove the pan from the heat and pour in the rum. Set the pan over high heat and carefully ignite the rum with a long match or grill lighter. Gently shake the pan until the flame subsides. Remove from the heat. Place a generous scoop of ice cream in 6 to 8 individual heatproof dessert bowls. Arrange 2 to 3 pieces of sautéed banana around each scoop of ice cream. Spoon on the sauce and serve immediately.