In the bowl of a stand mixer, add the flour, 0.5 cup sugar, and yeast. Stir to combine.
Scatter the softened butter over the flour mixture. In a small bowl, whisk together milk and eggs. Add the milk mixture, vanilla, and salt to the flour mixture.
Using the dough hook attachment, beat on low speed until a smooth and elastic dough forms, about 10 minutes.
Turn the dough out onto a work surface. Knead 4 to 5 times or just enough to shape it into a ball. Place in a lightly oiled bowl, cover, and let rise until doubled in size, about 1 to 1.5 hours.
Punch down the risen dough and turn it out onto a clean work surface. Roll the dough out to 0.5-inch thickness. Cut out rounds using a 3-inch cookie cutter.
Place the cut dough rounds on a parchment-lined baking sheet, cover, and let rise for 45 minutes.
Fill a large Dutch oven with oil 3 inches deep and heat to 350°F (175°C).
Fry the doughnuts in batches. Cook for 2.5 minutes, then turn over and cook an additional 2.5 minutes, or until each side is a deep golden brown.
Remove from oil and drain on a paper towel-lined plate. Roll each warm Italian doughnut in a bowl containing the remaining 0.25 cup of sugar.
Once cool, use the tip of a small knife to make a small incision on the side of each doughnut and fill with pastry cream or chocolate hazelnut spread.
Notes
Ensure your frying oil is consistently at 350°F to prevent greasy doughnuts.