Prep milk. Warm milk in the microwave on low power for 1 minute. (If using active, dry yeast, you can mix it with the milk at this time if you prefer, or add it with other ingredients.)
Mix all ingredients except flour. In a stand mixer with a paddle attachment or with an electric mixer with a large bowl, beat the milk, yeast, butter, sugar, eggs, salt, and vanilla.
Add flour. Continue to mix in flour one cup at a time until dough starts to pull from the sides of the bowl. Switch to a dough hook with a stand mixer and knead an additional 5-6 minutes or knead dough by hand.
First rise. Transfer dough to a greased large bowl, cover with greased plastic wrap, and let rise 1 to 1 ½ hours.
Roll out and cut. Roll dough onto a floured surface until it is approximately ½ inch thick. (If you don’t have a doughnut cutter, use a 3 ½ inch circular cutter to cut the outside circles and a 1-inch cutter in the center for the inside circles for the doughnut holes.) Cut the outside circles as close together as possible and place dough circles on two baking sheets lined with parchment paper. Combine and roll out scraps of dough until all the dough is used. The final doughnuts will not look quite as nice, but most imperfections will be covered by the icing.Tip: Use a sharp edge cutter, not glass. A dull glass cup, for example, will crimp the edges, and they won't rise as high. Second rise. Cover cookie sheets with a cotton towel and let rise for 1 to 1 ½ hours. 15 minutes before doughnuts are done rising, pour oil in a deep pot until about 2 inches deep. Bring the oil to a temperature of 360℉, using a thermometer to monitor the temperature.
Prep for cooling. Line a cookie sheet with parchment paper or a paper towel and place a cooling rack on top. Place cookie sheet near your pan with oil.
Fry. Cut parchment paper under risen doughnuts into squares. Using a spatula, lift the doughnut with the parchment paper underneath into hot oil. Place two doughnuts or several doughnut holes in the oil to cook at once. Cook doughnuts for one minute on one side, flip using a strainer spoon or slotted spoon, and cook for 45 seconds on the other side. Doughnut holes cook a little faster. Remove the parchment paper with tongs and discard while the doughnuts cook; it will just float and is safe in the oil. Place the cooked doughnuts onto the prepared cooling rack using a strainer spoon. Cook all the doughnuts and let them cool. While they cool, prepare the icing.