Place the warm water in a large mixing bowl. Stir in the 1 tablespoon granulated sugar until it dissolves. Sprinkle the yeast in the water and let mixture stand for 5 minutes or until it bubbles. Stir in the 3/4 cup flour to make a smooth, wet batter. Cover and let batter rise in a warm place for 30 minutes.
In another small bowl, beat the egg, egg yolk, the 1/2 cup granulated sugar, and salt with a wooden spoon. Pour egg mixture into the yeast mixture. Stir in 3 cups of the flour, the butter, dark rum, orange peel, and lemon peel. Stir until combined. Stir in as much of the remaining 1/2 cup flour as you can.
Transer dough to a lightly floured surface. Knead in remaining flour to make a moderately soft dough (3 to 4 minutes). Shape dough into a ball. Place ball in a lightly buttered bowl, turning once to grease the surface of the dough. Cover dough and let rise in a warm place until double in size (1-1/4 to 1-1/2 hours).
Punch down dough. Transfer dough to a lightly floured surface; divide in half. Cover; let rest for 10 minutes. With a dough scraper or sharp knife, chop each dough portion into 15 pieces. To shape, roll each piece of dough into a rope about 1/2 inch thick and about 8 inches long. Form each rope into a ring and pinch the ends together. Arrange the rings 2 inches apart on a baking sheet lined with parchment paper or a sheet greased with butter. Cover the sheets with kitchen towels and let the bussolai rise for 30 minutes. To bake two sheets of bussolai at a time, place a rack in the upper third of the oven and another in the lower third. Preheat oven to 375 degrees F.
Bake the bussolai in preheated oven for 12 to 15 minutes or until golden brown. If baking two baking sheets at once, rotate the pans halfway through baking. Transfer bussolai to a wire rack and cool completely.
Meanwhile, for the glaze, place the lemon and/or orange juice in a large bowl. Sift the powdered sugar into the juice. Stir to combine. (If you prefer a thinner glaze, stir additional juice into the glaze.)
Dip top of each ring into the glaze and flip it upright and place on a wire rack. Let the glaze set. Arrange bussolai on a serving platter or hang them on the Christmas tree. Makes 30 servings.