Proof the yeast by adding it to the milk and a pinch or two of sugar in your stand mixer. Let it rest for 10 minutes. It should form a frothy foam on the milk. If this does not happen, you must start over with new yeast.
Mix together the flour, sugar, and salt together, then add it to the stand mixer and beat on low until a dough forms. Add the eggs one at a time waiting until each is completely incorporated before adding the next. Add in the butter a tablespoon at a time waiting until the butter is incorporated before adding more.
Use a dough hook to knead the dough for about 10 minutes until the dough forms a soft, silky ball.
Grease a bowl lightly and place the dough in. Cover with plastic wrap. Allow to rise for 1 hour.
Punch down the dough and roll out until about ½ inch thick. Use a doughnut cutter or mouth of a cup to cut out doughnuts that are about 3 ½ inches in diameter. Re-roll any excess dough and cut.
Place the doughnuts on a greased sheet of parchment paper. Cover with plastic wrap and allow it to rise for another hour.
Heat at least 3 inches of oil to 350F. Fry each doughnut until a golden brown - about 2 - 3 minutes on each side. You should be able to fit roughly 2 - 3ndoughnuts in a pan or deep fryer depending on the size. Just make sure the doughnuts do not touch while frying.
Allow the doughnuts to drain briefly on a paper towel-lined wire rack. Toss in granulated sugar
Fit a piping bag with a small round piping tip and then will with the jelly. Insert the piping tip into the donut and fill with a few tablespoons of the jelly.