Step 1

Whisk together flours. Combine water and milk in the bowl of an electric mixer fitted with the dough-hook attachment, then sprinkle yeast over top. Stir in 1 cup flour mixture. Cover with plastic wrap; let stand in a warm place until bubbling and doubled in volume, about 1 hour. Add egg and yolk, granulated sugar, salt, and remaining flour mixture. Beat on medium speed 3 minutes. Continue beating while adding butter, a few pieces at a time, beating to combine after each addition. When all butter has been added, keep beating until a shiny, sticky dough forms around hook, about 4 minutes more. Scrape down sides of bowl.

Step 2

Cover dough with plastic wrap. Let stand in a warm place until almost doubled in volume, about 1 hour. Refrigerate at least 8 hours and up to 24.

Step 3

Cut out 9 three-inch squares of parchment. Spray a baking sheet with oil; arrange parchment squares in a single layer. Spray parchment. Transfer dough to a lightly floured baking sheet; pat out 1/2 inch thick. Using a floured 2 3/4-inch round cutter, cut out about 9 rounds. Transfer to parchment squares. Drape with oil-sprayed plastic wrap and let stand in a warm spot until slightly more than doubled in bulk and very soft, about 1 hour.

Step 4

Meanwhile, pour oil into a large, deep, heavy pot, such as a Dutch oven, until it reaches 2 inches up side, leaving about 2 1/2 inches headroom. Clip a deep- fry thermometer to pot and heat oil over medium to 350 degrees to 360 degrees. Working in batches of 3 so as not to crowd pot, carefully use parchment to transfer donuts to hot oil and cook, flipping a few times, until puffed and golden brown all over, 2 to 3 minutes total for donuts (1 1/2 to 2 minutes for holes), maintaining oil temperature between 350 degrees and 360 degrees at all times. Transfer donuts to a wire rack set in a rimmed baking sheet; let cool completely.

Step 5

Fill a pastry bag fitted with a coupler and a Bismarck piping tip (such as Ateco #with jelly. Fill each donut with about 2 teaspoons jelly (it should puff slightly in center). Dust with confectioners' sugar.