Tip: to make your own superfine sugar, place the sugar in a clean coffee mill or mini-chopper. Blend until crystals are fine but still "sparkle". Be sure not to blend too long, or the sugar will turn to powdered sugar.
Over low heat in a large, heavy pan, combine the sugar, rum, butter, lime zest and lime juice.
Cook for a few minutes, stirring often to avoid burning. Cook until the sugar has dissolved.
Add the bananas to the pan, turning to coat them with the sugar/rum mixture.
Increase heat to medium, and cook the bananas for about 5 minutes or until the bananas are a beautiful golden color.
Remove bananas from pan and cut in half lengthwise.
For each serving, place two banana slices on a dessert plate. Serve with a scoop of vanilla ice cream and generous drizzle of the hot sugar/rum sauce.