Lightly oil a large bowl and set it aside. In the bowl of a stand mixer, combine the flour, 1/2 cup of the sugar, and the yeast. Scatter the softened butter over the top. In a small bowl, whisk the warm milk and eggs together, then add them to the flour mixture along with the vanilla and salt.

Fit the mixer with the dough hook and mix on low for about 10 minutes, until the dough becomes smooth and elastic. It should feel tacky, but it should not stick to a clean finger. Turn the dough out onto an unfloured surface and knead it 4 to 5 times into a ball. Place it in the oiled bowl, turn to coat, cover, and let it rise in a warm spot for about 1 1/2 hours, until doubled.

While the dough rises, cut twelve 4-inch squares of parchment and set them on a baking sheet. After the first rise, punch down the dough, knead it 4 to 5 times, and roll it out to a rectangle about 3/4 inch thick. Cut 12 rounds with a 3-inch cutter, rerolling scraps as needed. Set each round on a parchment square, cover loosely, and let them rise again in a warm spot for about 1 hour, until doubled.

Pour oil to a depth of 3 inches into a large Dutch oven. Clip on a deep-fry thermometer and heat the oil over medium heat to 300 to 310°F. Line a rimmed baking sheet with paper towels, and put the remaining 1/4 cup sugar in a medium bowl. Working 2 or 3 at a time, lift each round by its parchment and lower it dough-side down into the oil. Use tongs to pull the parchment away.

Fry the bomboloni for 2 1/2 minutes, then flip and fry another 2 1/2 minutes, until deep golden brown on both sides. Adjust the heat as needed to keep the oil between 300 and 310°F. Lift them out with a slotted spoon or spider and drain on the paper towels. Let them cool for 2 minutes, then roll each warm doughnut in the sugar. Set them on a wire rack to cool completely. Once cool, cut a small slit in the side of each doughnut and fill with pastry cream.