In a large bowl, combine the warm milk, 1 tablespoon of granulated sugar (taken from the 0.5 cup total), and the active dry yeast. Stir gently and let sit for 5-10 minutes until the mixture becomes foamy, indicating the yeast is active.

Add the remaining granulated sugar, large eggs, softened unsalted butter, salt, and vanilla extract to the yeast mixture. Whisk well to combine thoroughly.

Gradually add the all-purpose flour, mixing with a wooden spoon or stand mixer (with a dough hook) until a shaggy dough forms. It will be sticky initially.

Turn the dough out onto a lightly floured surface. Knead for 8-10 minutes, either by hand or with a stand mixer, until the dough is smooth, elastic, and no longer sticky.

Lightly grease a clean bowl with a little oil. Place the dough in the bowl, turning once to coat. Cover the bowl tightly with plastic wrap or a damp kitchen towel and let it rise in a warm place for 1 to 1.5 hours, or until it has doubled in size.

Once the dough has risen, gently punch it down to release the air. Turn it out onto a lightly floured surface and roll it out evenly to about 0.5 inch (1.25 cm) thickness.

Using a 2.5 to 3-inch (6-7.5 cm) round cookie cutter, cut out individual bomboloni. Gather and re-roll any dough scraps to cut additional doughnuts until all the dough is used.

Carefully transfer the cut doughnuts to parchment-lined baking sheets, ensuring to leave about 2 inches of space between each. Loosely cover them with plastic wrap or a clean kitchen towel and let them rise for a second time for 30-45 minutes, or until visibly puffy.

While the doughnuts are undergoing their second rise, heat vegetable oil in a large, heavy-bottomed pot or Dutch oven over medium heat to 350°F (175°C). Use a kitchen thermometer to accurately monitor the oil temperature.

Carefully place 2-3 bomboloni into the hot oil, ensuring not to overcrowd the pot, which can lower the oil temperature. Fry for 2-3 minutes per side, until they are beautifully golden brown and puffed up.

Using a slotted spoon or spider, remove the fried bomboloni from the oil and place them on a wire rack lined with paper towels to drain any excess oil.

While the bomboloni are still warm, roll them generously in granulated sugar until fully coated on all sides.

If desired, once cooled slightly, use a pastry bag fitted with a long, thin tip to inject pastry cream, your favorite jam, or Nutella into the center of each bomboloni.

Serve warm and enjoy your delightful homemade Italian bomboloni!