Cover a small rimmed baking sheet (one that will fit in your freezer) with parchment paper, and place in the freezer for 10 minutes. When chilled, remove pan from freezer. Quickly scoop ice cream into 12 small balls, and place on chilled parchment-lined sheet. Keep in freezer until firm and ready to serve, at least 1 hour and up to 1 day ahead.
When ready to serve, peel bananas and quarter them, cutting lengthwise and then crosswise; set aside. Pour banana liqueur and dark rum into separate glass measuring cups; set aside. Heat 3 tablespoons butter in a large skillet over medium heat. Sprinkle sugar and cinnamon over butter, and cook until sugar is dissolved.
Remove pan from heat, and carefully stir in banana liqueur. Add bananas, flat side down, and cook until softened and lightly browned on the bottom.
Remove pan from heat, and add rum. Return to heat, and cook about 10 seconds to allow rum to heat up. If using a gas stove, carefully tip the pan away from you until the vapors from the rum ignite. (Alternatively, light the rum with a long match.) When the flames have subsided, remove pan from heat and gently stir in remaining 3 tablespoons butter.
Place 3 scoops of ice cream in each of four serving bowls. Spoon the banana mixture and the sauce over each. Serve immediately.