Stir together yeast and milk in bowl of mixer and let stand until it appears creamy, about 5 minutes. (If mixture doesn't appear creamy, start over with new yeast.)
Mix in 1/2 cup flour at low speed. Cover bowl with plastic wrap and a kitchen towel and let rise in a draft-free place at warm room temperature until doubled, about 1 hour.
Mix in egg, butter, sugar, vanilla, cinnamon, salt, and remaining 1/2 cup plus 2 tablespoons flour at low speed until combined. Increase speed to medium and beat dough until smooth and elastic, 5 to 7 minutes.
Scrape dough into center of bowl and dust with additional flour. Cover bowl with plastic wrap and a kitchen towel and let dough rise in a draft-free place at warm room temperature until doubled, about 1 hour.
Punch down dough and turn out onto a well-floured surface (dough will be very sticky.)
Cut into 16 equal pieces and form into balls with floured hands.
Transfer to a lightly floured baking sheet 1 inch apart.
Heat 2 1/2 inches oil to 350°F in a 4- to 5-quart pot over medium heat. Fry bomboloni in batches of 4, turning frequently with tongs for even browning, until puffed and golden brown, 2 to 3 minutes per batch.
Transfer with a slotted spoon to towels to drain. Return oil to 350°F between batches.
Cool bomboloni to warm and serve with sauces.
•Bomboloni can be formed, but not fried, 1 day ahead and chilled on a rimmed sheet pan, loosely covered with plastic wrap.
Let stand at room temperature 20 minutes before frying.•Although bomboloni are best freshly made, they can be fried a day ahead, cooled, and kept covered with plastic wrap at room temperature. Reheat, uncovered, in a sheet pan on a rack in a 300°F oven until warm, 4 to 5 minutes.•Oil can be cooled to warm, strained through a paper towel–lined sieve, and reused once more.
