Make sure milk is nice and warm, but not hot. Add sugar to milk and stir to dissolve. Add yeast, stir gently and then let sit for 10 minutes.
Melt butter in separate bowl until butter is almost melted. Stir to finish melting so butter won’t be overly hot.
Add beaten eggs to melted butter, stirring constantly to make sure the butter’s not too hot for the eggs.
Add the egg/butter mixtures and using a dough hook pour in the yeast mixture. Allow the dough hook to stir a couple of minutes, making sure it’s thoroughly combined.
With the mixer still going, add helpings of the flour mixture in 1/4 to 1/2 cup increments until all the flour is gone.
Stop mixer, scrape bowl, then mix on medium speed for 5 minutes. Stop mixer, scrape bottom of bowl, turn back on for 1 minute more. Turn off mixer and let bowl sit 10 minutes.
After 10 minutes, transfer dough to lightly oiled bowl. Toss the dough to coat, cover with plastic wrap and refrigerate for at least 8 hours, or overnight.
Make Doughnuts
Turn out dough onto lightly floured surface. Roll out to 1/4 to 1/3 inch thickness. Using a 3 inch doughnut cutter, cut as many as you can, then rollout remaining dough and cut as many as you can.
Using a 1 1/2 inch cutter, cut holes out of doughnuts and remaining dough. Place on floured baking sheet. Cover with a large tea towel and place in a warm place in your kitchen.
Allow doughnuts to raise for at least an hour. Doughnuts should be visibly puffier and appear to be airy.
To Fry Doughnuts
Heat plenty of vegetable oil in a large pot until temperature reaches 375 degrees. Keep the thermometer in the pan to continually monitor.
Gently grab doughnuts one or two at a time and ease them into the oil. Allow them to cook 1 minute on each side, they will brown very quickly.
Remove doughnuts from oil with a slotted spoon, allowing oil to drip off. Drain on several layers of paper towels. Count to five and flip over on more paper towels. Do that one more time. Allow to cool slightly
To Glaze
Mix all ingredients until completely smooth. One by one dip doughnuts into glazed until halfway submerged. Remove from glaze and set on cooling rack over cookie sheet. Serve warm if possible or at room temperature.