Step 1

For the doughnuts: Warm the milk to 100°F to 110°F. Add the sugar to the milk and stir to dissolve. Add the yeast and stir gently, then let the mixture sit for 10 minutes.

Step 2

To a separate bowl, add the butter and microwave until almost melted. Stir to finish melting the butter, so it won't be overly hot. Slowly add the beaten eggs to the melted butter, stirring constantly.

Step 3

To the bowl of an electric mixer fitted with the dough hook, add the egg mixture. With the mixer on medium-low speed, pour in the yeast mixture. Allow the dough hook to stir this mixture for a couple of minutes, making sure it's thoroughly combined. Meanwhile, in a small bowl, whisk together the flour and salt.

Step 4

With the mixer still going, add the flour mixture in 1/2-cup increments, mixing until well combined after each addition, until all the flour is gone. Stop the mixer, scrape the bowl, then turn the mixer on the same speed for 5 to 7 minutes, stopping to scrape the sides and bottom of the bowl once.

Step 5

Transfer the dough to a lightly greased bowl. Toss the dough to coat in oil, then cover the bowl with plastic wrap and place straight in the fridge. Refrigerate the dough for at least 8 hours, or overnight.

Step 6

To form the doughnuts, turn out dough onto a lightly floured surface. Roll it out to 1/4- to 1/3-inch thickness. Using a 3-inch round cutter, cut as many rounds as you can. (You can re-roll the remaining dough and cut as many as you can. Let the dough rest for 15 minutes, before re-rolling.)

Step 7

Cut holes out of each round using a 1-inch round cutter. Place both doughnuts and holes on a floured baking sheet. Cover with plastic wrap and place in a warm place in your kitchen. (My kitchen is very drafty, so I have to briefly warm the griddle, then turn it off and set the sheets on top to keep warm. Allow doughnuts to rise undisturbed for at least 1 hour or 1 hour 15 minutes, if necessary. The doughnuts should be visibly puffier and appear to be airy.

Step 8

Meanwhile, heat plenty of vegetable oil in a large pot until the temperature reaches 375°F. Keep the thermometer in the pan to continually monitor.

Step 9

Working one to two at a time, gently grab the doughnuts and ease them into the hot oil. Allow them to cook 1 minute on each side; they will brown very quickly. Remove the doughnuts from the oil with a slotted spoon, allowing all of the oil to drip off. Immediately place the doughnuts on several layers of paper towels. Count to five, then flip it over onto a clean part of the paper towels. Count to five, then flip it over again—the purpose, obviously, is to drain as much grease as possible before it soaks into the doughnut. Repeat with remaining doughnuts and holes. The holes will cook more quickly than the doughnuts; about 30 seconds per side. Allow the doughnuts to slightly cool.

Step 10

For the glaze:  In a medium bowl, whisk together the confectioner's sugar, salt, vanilla, and water or milk until completely smooth. One by one, dip the doughnuts into the glaze until halfway submerged. (Note: completely submerge doughnut holes, then remove with slotted spoon.)  Remove from the glaze, then turn right-side up on a cooling rack over a cookie sheet (to catch dripping glaze.) Serve warm if possible, or at room temperature.