Lightly oil a bowl. Heat milk in a small saucepan over medium heat until mixture reads between 105°F and 110°F on an instant-read thermometer. Stir in yeast and granulated sugar. Set aside until foamy, 4 to 6 minutes.
Whisk together yeast mixture and eggs. Add butter and salt and whisk to combine (some lumps of butter may remain). Gradually add flour, stirring until a dough forms. Beat with a dough hook attached to an electric mixer on low speed, 4 to 6 minutes (dough will be sticky).
Transfer mixture to prepared bowl. Cover with plastic wrap and let sit until doubled in size, 1 to 1 1/2 hours or overnight in refrigerator (if overnight then let come to room temperature before rolling, about 1 hour).
Lightly oil a large baking sheet. Turn dough out on a lightly floured work surface; lightly dust top. Press or roll to 1/2-inch thickness. Cut dough with a floured 1-inch round cutter; transfer to prepared baking sheet. Re-roll and cut scraps. Cover with a towel; let rise until puffed, 15 to 20 minutes.
Place a wire rack on a rimmed baking sheet. Heat oil in a large, deep saucepan to 350°F. Working with 3 or 4 donuts at a time, cook, turning once, until golden brown, 2 to 3 minutes. Use a large slotted spoon to transfer donuts to cooling rack; let cool.
Place confectioners’ sugar in a bowl. Add donuts and toss to coat; return to wire rack.
Cut a narrow slit into the side of each donut. Fill a large piping bag, fitted with 1/4-inch round tip, with half of jelly. Fill donuts until jelly just starts to come out of hole. Repeat with remaining jelly and donuts.