Heat milk in the microwave for 1 minute. Stir and test temperature. Your milk should be warmer than lukewarm, but not hot. If required, heat for an extra 25 seconds. Add 1 tablespoon sugar and sprinkle with yeast. Stir gently and set aside for 10 minutes. After 10 minutes, your yeast mixture should have grown and become frothy. If it has not grown, your yeast is dead and you must start over with either a new batch of yeast or perhaps your milk temperature was too hot and it accidentally killed the yeast.
In the bowl of your stand mixer, using the paddle attachment, cream the butter with 1/4 cup granulated sugar. Add eggs, one at a time, scraping sides of the bowl as needed. Add salt.
Switch your attachment to the dough hook. Alternating between the flour (total of 5 cups) and yeast mixture, add them both to the bowl, starting and ending with flour. Mix well after each addition. Mix with the dough hook for several minutes until the dough pulls away from the sides of the bowl. If dough does not pull away from the sides, add the additional 1/4 cup of flour.
Transfer the dough to a lightly greased bowl. Turn your oven on and let the oven begin to heat up for 3 or 4 minutes then turn the heat off. Warming the oven this way will give your dough excellent conditions for rising. Cover the bowl with a tea towel and place in the preheated oven. Do not turn the oven back on. Insert a wooden spoon or similar object in the oven door to prevent it from closing all the way. This will allow a small amount of the heat to escape. You don’t want to cook the dough yet, just warm it enough for it to rise. Let the dough rise for one hour or until it has doubled in size.
Transfer the dough to a well floured surface and roll the dough to 1/2″ thickness. Use a donut cutter to cut the donuts and donut holes. If you don’t have a donut cutter, you can use two different sizes of circle shapes ie: drinking glass and bottle top to make your donuts and holes.
Transfer the donuts and donut holes to a parchment lined cookie sheet, cover with a towel, and place back in the oven to rise for an additional 30 minutes.
Pour oil into a medium-sized, high sided pot. For safety sake, you do not want your oil to be more than 2″ deep, and definitely not more than halfway up the side of the pot. Heat oil to 375 degrees. A candy thermometer comes in handy here, but if you don’t have one, heat the oil on medium-high heat for about 5 – 7 minutes. Test your oil temperature with one of the donut holes. If it browns too quickly, turn the heat down to medium and wait a minute or two to start frying.
Carefully remove the donuts from the cookie sheet and drop one or two at a time into the hot oil. Cook for 45 seconds to 1 minute, or until browned, and carefully turn over using a slotted spoon, tongs, or two chopsticks also works very well. Cook on the other side for 45 seconds to one minute. Remove from the oil and place the donuts on a paper towel lined baking sheet to drain.
Let the donuts cool slightly before coating with cinnamon sugar mixture or glaze.
For The Sugar Cinnamon
In a bowl, mix the sugar and cinnamon together until well combined. Gently roll the fried donuts and holes to coat.
For The Glaze
In a bowl whisk all ingredients until smooth. For a thicker glaze, use 1/4 cup of milk, for a thinner glaze, use 1/2 cup of milk. (Pictured with a thinner glaze.) Place the fried donut in the bowl of glaze and flip to coat both sides. Once coated with glaze, let the donuts sit on a wire cooling rack over parchment paper or a baking sheet, to catch the dripping glaze, until they are dry.