In a medium saucepan, melt the butter. Add the brown sugar. Cook together, stirring, until the sugar melts and starts to caramelize. Add the bananas, tossing to coat in the caramel. Add the cinnamon and vanilla extract. Cook for 2 to 3 minutes, until the bananas start to soften. Remove the bananas and divide among 4 serving plates.
Add the rum to the saucepan and turn off the heat. Using a long-reach butane lighter, carefully flambé, or ignite, the sauce. Let the flame burn and go out. Drizzle the sauce over the bananas and serve.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.