In a large bowl or the bowl of a stand mixer fitted with the dough hook, stir together 2¾ cups of the flour, the nonfat dry milk powder, sugar, salt, and yeast until combined. Add the eggs, egg yolk, warm water, and butter and stir until a dough forms.

On a lightly floured surface, knead the dough by hand for 10 to 15 minutes, adding just enough of the remaining ¼ cup flour until the dough is soft, smooth, and elastic. Alternatively, knead in the stand mixer on low speed for 10 minutes, adding flour as needed.

Shape the dough into a ball and place it in a large lightly greased bowl, turning once to coat. Cover with plastic wrap and let rise in a warm place for 1 hour, then transfer to the refrigerator and chill for 8 hours or overnight, no longer than 12 hours.

Punch down the risen dough. On a lightly floured surface, press the dough into a ½-inch thick circle and cut out rounds using a 3-inch round cookie cutter. You may re-press the scraps once to cut more doughnuts. Place on a parchment-lined baking sheet, cover with a towel, and let rise for 1 hour.

Heat 2 inches of vegetable oil in a large saucepan over medium heat until a candy thermometer reads 350°F. Fry 2 to 3 doughnuts at a time, turning once, until deep golden brown on both sides. Transfer to a paper towel-lined plate and repeat with the remaining doughnuts.

Use a #230 round decorating tip to pipe blackberry jam into the center of each doughnut. Dust generously with powdered sugar and serve.