In the bowl of a mixer fitted with a dough hook, combine the eggs and the sugar on low speed, adding slowly the rum, the melted butter and the flour sifted with the yeast. Once the dough has come together, put it on a pastry board and use your hands to knead it. Put the dough in a warm place for 2 hours.
Shape small portions of the dough into long “snakes”. On a floured surface, cut snakes into half-inch pieces.
Heat the oil and keep it at the right frying temperature without over-heating it. Drop into the oil and when they have puffed up, remove them and drain on absorbent paper.
Arrange the “castagnole” on a serving plate and sprinkle with confectioners sugar.