Peel the bananas. Slice them at an angle into pieces 1-inch thick.
Heat a large sauté pan over medium-high heat. Melt the butter. When the butter foams, lay in the pieces of banana, cut-side down, and the cinnamon. Let the bananas cook about 30 seconds. Add the brown sugar stirring gently, turning the banana pieces once, until the sugar melts.
Turn bananas once more and add the banana liqueur, shaking and swirling the pan to combine the sauce ingredients. Add the rum and ignite either by tipping the pan towards the flame or by lighting contents with a match. Remove pan from burner and allow the flames to burn out completely. Place the pan back on the burner, turn up the heat and let the sauce reduce, spooning over bananas.