3/4 teaspoon active dry yeast (from a 1/4-ounce package)
6 tablespoons warm whole milk (105-115°F)
1 cup plus 2 tablespoons all-purpose flour, plus more for dusting
1 large egg
1 1/2 tablespoons unsalted butter, softened
1 1/2 tablespoons sugar
1/2 teaspoon vanilla extract
1/4 teaspoon ground cinnamon
1/4 teaspoon salt
About 2 quarts vegetable oil for frying
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.