In a small bowl, dissolve the yeast in room-temperature warm water. Let it sit for about 10 minutes.
In a saucepan, heat the milk until it's hot but not boiling. Remove from heat and allow it to cool.
In a mixing bowl, combine the sugar, shortening, and salt. Mix them together using a paddle attachment for about 2 minutes.
Pour in the cooled milk and let it come to room temperature.
Add the egg and vanilla extract to the mixture. Then add the yeast-water mixture.
Switch to a dough hook attachment and gradually add the flour. Add a little at a time until the dough forms into a ball. You may need more or less flour.
Transfer the dough to a floured surface and knead it for about 5 minutes.
Place the dough in an oiled bowl and let it double in size, which should take about 1 hour.
On the floured surface, roll out the dough to a thickness of 1/2 inch. Use a cookie cutter to cut out donut shapes, and make a hole in the center of each donut.
Place the cut-out donuts on a parchment paper-lined tray.