In a large mixing bowl, stir with a stand or handheld mixer fitted with a whisk attachment milk, sugar, and yeast to combine. Let sit for about 5-10 minutes until it starts to get foamy on the surface.
Preheat oven to 200°F (100°C). Line 3 baking sheets with parchment paper. Set aside.
Add eggs, butter, and salt and whisk to combine.
Replace whisk attachment and attach a dough hook. Add 3+¾ cups flour and knead on medium-high speed until the dough comes together about 4-5 minutes. The dough should be slightly sticky to touch and should start to come off the sides of the bowl. If it doesn't come off the sides of the bowl at this stage, turn the speed down to low and add 1 tablespoon of flour at the time (up to 4 cups in total) and mix on low speed until you reach this point. But don't add too much flour or your doughnuts end up dry. How much flour you need depends on the humidity and temperature where you live. Make sure that you stay at around 4 cups highest. (Watch the video and the step-by-step photos to see the required consistency.) Once you reached the required consistency, turn on medium-high speed and knead for about 5-10 minutes.
Transfer dough to a lightly greased bowl (brush a few drops of oil in a bowl or use a non-stick spray) and lightly grease the top of the dough.
Turn off preheated oven and place bowl in the oven to rest for about 10-15 minutes. Leave the oven door ajar.
Remove bowl from the oven and preheat again to 200°F (100°C).
On a very lightly floured surface, roll out the dough, about ½ inch (1,2 cm) thick. Take as little flour as possible. Just enough to roll out the dough without any problems. Cut out 3-inch (7,5cm) donuts with a lightly floured donut cutter. Place on prepared baking sheets. Turn off preheated oven and place all baking sheets in the oven to rise for about 30 minutes until they are doubled in size. Leave the oven door ajar.
In a heavy-bottomed saucepan, heat oil to 360°F-370°F (180°C-185°C). Use a thermometer. Fry doughnut rings on each side for about 60-90 seconds until golden brown. Don't crowd the pan too much. I fried just one or maximum two doughnuts at the same time. Fry doughnut holes for about 30 seconds on each side. Place fried doughnut rings and holes on a wire rack or paper towels.
In a large mixing bowl whisk butter, vanilla, and powdered sugar to combine. Add one tablespoon of hot water at the time and stir to combine. The consistency should be thick but liquid. When the glaze is too thick, it will be too much glaze on the doughnuts, and when it is too thin, it will run down the doughnuts. (Watch the video and the step-by-step photos to see the required consistency.) Dip warm doughnuts on each side into the glaze and let drip off any excess glaze. Place on a parchment paper layered baking sheets or wire rack to dry. The glaze will dry and harden after about 20-30 minutes.
Serve immediately when the glaze is dried. Although doughnuts are best eaten fresh, you can store them in the fridge in an airtight container up to 1 day. Let come to room temperature before serving.