Prep milk. Warm milk in microwave on low power for 1 minutes. (If using active dry yeast, you can mix it with the milk at this time if you prefer, or add 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 plastic wrap sprayed with butter spray, and let rise 1 to 1 ½ hours.
Roll out and cut. Roll dough onto a floured surface until dough 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 circles of dough on two baking sheets lined with parchment paper. Combine and roll out scraps of dough until all dough is used. The final doughnuts will not look quite as nice, but most imperfections will be covered by the icing.
Second rise. Cover cookie sheets with a cotton towel and let rise for 1 to ½ hours. 15 minutes before doughnuts are done rising, pour oil in a deep pot until about 2 inches deep. Bring oil to a temperature of 360 degrees, using a thermometer to monitor 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 donuts in squares. Using a spatula, lift the donut with the parchment paper underneath into hot oil. Place two donuts or several donut holes in the oil to cook at once. Cook donuts for one minute on one side, flip using a strainer spoon or slotted spoon, and cook for 45 seconds on the other side. Donut holes cook a little faster. Remove the parchment paper with tongs and discard while the donuts cook, it will just float and is safe in the oil. Use a strainer spoon to place the cooked donuts onto the prepared cooling rack. Cook all the donuts and let cool. Prepare icing while donuts cool.