Heat butter and brown sugar in a skillet over medium-high heat until butter melts and mixture begins to bubble, 3 to 5 minutes.

Place bananas, cut-side down, into the brown sugar mixture. Cook until just golden, about 30 seconds; flip and cook until golden on the other side, about 30 seconds. Turn off the heat.

With the heat off, pour rum and banana liqueur into the skillet. Turn the heat to medium-high or high; cook until alcohol ignites and flames, 1 to 3 minutes. Carefully add cinnamon. When the flames die, remove from the heat.

Scoop vanilla ice cream into two bowls. Divide banana mixture between the bowls and garnish with mint sprigs.