Melt the butter in a medium nonstick skillet over medium heat. Add the Dulce de Leche and mix with a heatproof rubber spatula until well combined. Cook for 3 minutes, stirring often. Add the banana slices, stir to coat them, and cook for 3 more minutes. Take the pan off the heat, stir in the rum, and return the pan to the heat. Cook for 2 more minutes, stirring well to coat the bananas with the sauce.
Transfer the bananas and sauce to plates, or spoon them over vanilla ice cream. Garnish with fresh mint leaves, if desired.
Dulce de Leche
Pour the milk into the top of a double boiler and put it over a pot of barely simmering water. Keep the water gently simmering for 2 hours, adding more hot water as the water evaporates. Start stirring once the milk begins to thicken, once it does, after about 30 minutes, stir every 15 minutes or so.
Store in an airtight container in the refrigerator for up to a month.