In a bowl, whisk together the egg yolks, milk, and butter. Add the flour, salt, and the sugar to the egg mixture, combine well, beat until mixture is smooth, and allow to rest in the refrigerator for about 2 hours.
In a dry clean bowl whip the egg whites until stiff, but not dry. Fold the egg whites into the batter with a rubber spatula.
In a large heavy-bottom saucepan, pour enough oil to fill the pan about a third of the way, about 3 to 5 inches.
Heat over medium heat until a deep-frying thermometer inserted in the oil reaches 350 degrees F. dredge the banana pieces in the confectioners' sugar then dip the pieces into the batter. Deep-fry the bananas in batches until golden brown, about 3 to 4 minutes.
Drizzle with chocolate sauce. Garnish with mint leaves.
In a heavy bottom saucepan, put chocolate and water. Over low heat, melt the chocolate, while stirring. Boil the mixture for 1 minute. Remove from the heat and add the rum.