1For the donuts: Add the granulated sugar to the warm milk in a medium bowl, and then add the yeast. Allow it to sit until the yeast starts to bubble, 5 to 10 minutes.
2In a small bowl, beat the eggs and then pour them into a bowl with the melted butter, whisking constantly. Add the butter/egg mixture to the bowl of an electric mixer. Pour in the milk/sugar/yeast mixture. With the hook attachment, turn the mixer to low speed.
3Mix the flour and salt in a bowl, and then retrieve ½ cup at a time and add it to the mixing bowl, allowing it to slowly incorporate into the liquid mixture.
4Continue mixing for 5 minutes after the flour is combined. Stop the mixer, scrape the bowl and then mix about 30 seconds more. Then place the dough in a lightly-oiled bowl, cover it with plastic wrap and refrigerate, 8 to 12 hours.
5Continue mixing for 5 minutes after the flour is combined. Stop the mixer, scrape the bowl and then mix about 30 seconds more. Then place the dough in a lightly-oiled bowl, cover it with plastic wrap and refrigerate, 8 to 12 hours.
6Turn out the dough on to a floured surface and roll it out to about ¼-inch thick. Use a donut cutter (or 2 concentric cutters) to cut out the donuts. Remove the holes and transfer the donuts to a lightly-floured baking sheet lined with a baking mat or parchment. Then - and this is the vital part - cover the donuts lightly with tea towels and place it in a draft-free area, at least 1½ to 2 hours. It will take that long for them to rise. If they don't seem to be rising much, move the pan to a warm place.
7Melt the shortening in a pot over a medium-high heat until it reaches 350 degrees F on a candy thermometer. (Or you can drop one of the donut holes in the oil, if it sizzles and immediately rises to the surface, the oil is ready.) The donuts should immediately float to the top and puff up. Then use a metal spoon or spatula to carefully flip them over to the other side. Remove them from the oil as soon as they're golden brown on both sides (this should take less than 1 minute in total). At the end, drop in the donut holes and fry them until they are golden brown.
8Place the donuts on paper-towel-lined-plates to drain. Don't worry if they're a little imperfect; if your fingers leave impressions when you dropped them into the oil, that just means they were extra light and fluffy.
9For the glaze: To glaze the donuts, mix the powdered sugar, ½ cup cold water, salt and vanilla in a bowl until smooth. Drop in the donuts one at a time. Quickly turn them over then remove them. Place them on a rack so any excess glaze can drip off.
10To make the most of the glaze, set the holes underneath the donuts so they can catch the extra good stuff.
11Squeeze the plastic teeth shut and insert them into the center of the donuts.