In a large mixing bowl, combine all-purpose flour, granulated sugar, instant yeast, and salt. In a separate bowl, whisk together the warmed milk, melted butter, and eggs. Pour the wet ingredients into the dry mixture and stir until a rough dough forms.
Transfer the dough to a lightly floured surface and knead for about 8-10 minutes until smooth and elastic. If using a stand mixer with a dough hook, knead on medium speed for 6-7 minutes. The dough should be soft but not sticky.
Shape the dough into a ball and place it in a lightly greased bowl. Cover with a clean kitchen towel or plastic wrap and let it rise in a warm spot for about 1 to 1.5 hours, or until doubled in size.
Punch down the risen dough and roll it out to about 1/2-inch thickness on a lightly floured surface. Use a round cutter or glass to cut out circles. Place them on a baking sheet lined with parchment paper, cover, and let them rest for 20 minutes to puff up slightly.
Heat vegetable oil in a deep frying pan to 350°F (175°C). Carefully place a few dough rounds into the hot oil, frying them for about 2-3 minutes per side until golden brown. Avoid overcrowding the pan. Use a slotted spoon to transfer them to a cooling rack lined with paper towels to drain excess oil.
Once the bomboloni are cool enough to handle but still warm, use a piping bag fitted with a small round tip to fill each doughnut with Nutella. Insert the tip into the side or bottom of each bomboloni and squeeze gently until filled.
Generously dust the filled bomboloni with powdered sugar to create the iconic snowcap look. Serve immediately for the best flavor and texture.