Prepare the Dough: In a small bowl, dissolve the yeast in warm milk and let it sit for 5-10 minutes until frothy. In a large mixing bowl, combine the flour, sugar, and salt. Gradually add the eggs, vanilla, butter, and milk-yeast mixture to the dry ingredients. Mix until a soft, sticky dough forms. Transfer the dough to a floured surface and knead for 8-10 minutes until smooth and elastic. Place the dough in a lightly oiled bowl, cover with a damp cloth, and let it rise in a warm place for about 1-2 hours or until it doubles in size.

Shape and Fry: Once the dough has risen, punch it down and roll it out to about 1/2-inch thickness. Use a round cutter to cut out donut shapes, then place them on a baking sheet lined with parchment paper. Let them rise for another 30 minutes. Heat the oil in a deep fryer or heavy-bottomed pot to 350°F (175°C). Fry the donuts in batches, turning them once, until golden brown (about 1-2 minutes per side). Transfer to a wire rack to drain excess oil.

Fill and Finish: Once cooled, use a piping bag fitted with a small nozzle to inject your choice of filling into each donut. Roll the filled Bomboloni in granulated or confectioners sugar for a sweet finish.