In a large skillet, heat the butter, brown sugar, cinnamon, and salt over medium heat until sugar dissolves, about 3 minutes. Stirring frequently.
While the butter mixture is cooking, slice the bananas into 1/2-inch thick diagonal slices.
Once the butter mixture starts to bubble, add the bananas in a single layer and cook for 1 minute. Turn the bananas over and cook for 1 more minute.
Very carefully pour in the vanilla extract and the rum and then using a long match or grill lighter held just over the liquid, ignite the rum to flambé. Cook until the flame goes out, about 1 minute. Remove from heat.
Add ice cream to 6 bowls and evenly top with the bananas and rum sauce. Serve immediately.
Light brown sugar can be used in a pinch, but not granulated sugar.