Prep the yeast: Add the yeast and warm water to the bowl of your mixer and lightly stir. Let it sit for 5 to 10 minutes until foamy. If your yeast doesn't get foamy it means your donuts won't rise.
Combine the dough: Add in the milk, sugar, salt, eggs, shortening and 2 cups of the flour. With the mixer on low, use the dough hook attachment and mix for a couple minutes. Beat in remaining flour 1/2 cup at a time until the dough comes clean from the sides of the bowl. Should take about 5 minutes. The dough should be smooth and elastic.
Allow to rise: Place the dough into a large greased bowl and let it rise until doubled in size in a warm, draft free environment.
Assemble the donuts: Working with half the dough at a time, roll it to 1/2 inch thick on a floured surface. Dip your donuts cutter in flour and cut out a dozen donuts with each half of the dough. Reroll the leftover dough and cut out more donuts. Let the donuts rise until doubled in size.
Make the glaze: Add the corn syrup, the icing sugar and vanilla to a saucepan over medium-low heat and whisk until smooth. Remove from heat and whisk in the milk. The glaze should be thin but not watery.
Fry the donuts: In a large Dutch oven or deep saucepan, heat the vegetable oil to 375 degrees. Fry the donuts for about 90 seconds on each side. Drain on a cooling rack until they're all finished cooling.
Finish the donuts: Dip donuts into the glaze while still hot.