Bloom the yeast: Pour the warm milk into the bowl of a stand mixer. Sprinkle the yeast on top and stir to dissolve; let stand until foamy, about 5 minutes. Make the dough: Add the sugar, salt, vanilla and eggs to the bowl and whisk until blended. Add 2 cups flour and mix with the dough hook to combine, stopping to scrape down the bowl. (The dough will be sticky; if it's too wet, add the remaining 1/4 cup flour.) Add the butter and mix until smooth. Let it rise: Brush a large bowl with vegetable oil; add the dough and cover with plastic wrap. Let rise in the refrigerator, at least 4 hours or overnight.
Cut out the doughnuts: Line 2 baking sheets with parchment and dust with flour. Dust a clean surface with flour; turn out the dough onto the surface. Dust the dough with flour, then gently pat it out to make it easier to roll. Using a rolling pin, roll out the dough until about 1/2 inch thick. Cut the dough into 1 1/2-inch rounds with a cookie cutter; transfer to the prepared baking sheets. Reroll the scraps and cut out more rounds. Dust the dough rounds lightly with flour, cover with plastic wrap and let rest 1 hour in a warm place.
Fry them: Heat 2 to 3 inches vegetable oil in a large pot until a deep-fry thermometer registers 360 degrees F. Working in batches, fry the doughnuts, turning, until golden brown, 1 to 1 1/2 minutes total. Using tongs or a skimmer, remove to paper towels to drain.
Fill with jelly: Fit a pastry bag with a 1/8-inch-round tip; place the bag in a glass and cuff the open end over the glass. Fill the bag with jelly. Insert the pastry tip into the side of a doughnut and gently squeeze to fill. Repeat with the remaining doughnuts. Dust with confectioners' sugar and serve warm. Photograph by David A. Land
Recipe courtesy Chuck Hughes for Food Network Magazine