1 doughnut cutter, or 2 concentric cutters Candy thermometer
For the doughnuts: 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.
In 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.
Mix the flour and salt in a bowl, and then retrieve 1/2 cup at a time and add it to the mixing bowl, allowing it to slowly incorporate into the liquid mixture.
Continue 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.
The next morning, remove the dough from the fridge and allow it to come to room temperature and rise, 1 1/2 to 2 hours. Put the dough in a warm spot, if necessary, to facilitate rising.
Turn out the dough on to a floured surface and roll it out to about 1/4-inch thick. Use a doughnut cutter (or 2 concentric cutters) to cut out the doughnuts. Remove the holes and transfer the doughnuts to a lightly-floured baking sheet lined with a baking mat or parchment. Then - and this is the vital part - cover the doughnuts lightly with tea towels and place it in a draft-free area, at least 1 1/2 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.
Melt 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 doughnut holes in the oil, if it sizzles and immediately rises to the surface, the oil is ready.) The doughnuts 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 doughnut holes and fry them until they are golden brown.
Place the doughnuts 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.
For the glaze: To glaze the doughnuts, mix the powdered sugar, 1/2 cup cold water, salt and vanilla in a bowl until smooth. Drop in the doughnuts one at a time. Quickly turn them over then remove them. Place them on a rack so any excess glaze can drip off.
To make the most of the glaze, set the holes underneath the doughnuts so they can catch the extra good stuff.