Let the yeast sit in room-temperature warm water for about 10 minutes.
In a saucepan, heat milk to a warm, but not boiling temperature. Allow it to cool after being removed from the heat.
Mix sugar, shortening, and salt together in a mixing bowl using a paddle attachment for about 2 minutes.
Let the milk come to room temperature after it has been cooled.
After adding the yeast-water mixture, add the egg and vanilla extract.
Using a dough hook attachment, gradually add the flour. Adding a little at a time will help the dough form into a ball.
Make the dough by kneading it on a floured surface for about five minutes.
Let the dough double in size in an oiled bowl for about an hour.
Prepare a floured surface and roll out the dough to a thickness of 1/2 inch. Cut out donut shapes with a cookie cutter, and make a hole in the center.
Line a tray with parchment paper and place the cut-out donuts on it.