Warm milk to about 90°F. Mix the yeast into the milk and gently sir. Let the mixture sit. It's ready when it becomes foamy, which takes 5 - 10 minutes.
Start the dough.
Beat eggs, melted butter (make sure it's cooled), sugar, salt, and yeast mixture on medium speed. Add the flour while mixing slowly, about a cup at a time.
Once the dough is combined, knead it on a floured surface until smooth.
Once smooth, set it in an oiled bowl that is at least twice the size of the dough. Cover it with a towel and let it rise at room temperature for about an hour.
Cut the donuts.
Once your dough has rested for an hour, roll it out to about ½-inch thickness on a floured surface. Cut out donuts with concentric cookie cutters or a drinking glass for the large circle and a shot glass for the hole (this is what I did). Your donut cutter should be about 3" across. Cut until all dough is used.
Second rise.
Place your cut donuts and donut holes on two floured baking sheets, ensuring to evenly space them out. Cover again with a towel and let them rest in a warm place for about 45 minutes. They should puff up quite a bit (see my photo above).
Heat the oil.
When there's about 15 minutes left before the donuts are done rising, heat the oil in a heavy-bottomed pot such as a Dutch oven over medium heat.
Heat to 375°F. While it's heating, line a surface, plate, or cooling rack with paper towels so you have a place to drain your donuts when they're done frying.
Showtime.
Once your oil reaches 375°F, carefully place a few donuts into the oil using a metal utensil. After about a minute, flip them. You want them to be golden brown all over. Pay attention closely, as they will cook quite quickly, and the donut holes will cook even faster.
Once done, drain the donuts and donut holes on paper towels. Glaze and decorate as desired and enjoy.
To make the glaze:
Whisk powdered sugar, milk, and vanilla extract until smooth. Glaze donuts as desired.