In a small bowl, heat the milk and water to 115 degrees. Mix in the sugar and yeast, and set aside to bloom. If the yeast is active, the mixture will become foamy.
In the bowl of your stand mixer, beat the eggs and the yeast mixture together with a fork until just combined. Add in the bread flour, and 3/4 cup all purpose flour, and the salt.
Mix this together until a very sticky dough forms- it will still be sticking to the sides, but should be very thick. See photos above. Add more flour if the mixture is too wet- but not so much that it completely pulls away from the sides.
Using the dough hook attachment, knead the dough on low-medium speed for about 5-10 minutes, or until the dough starts to pull away from the sides of the bowl.
Add in the butter one tablespoon at a time, letting it knead into the dough before each addition. This process can take a while, and you may need to scrape down the sides of the bowl some to ensure that the butter gets incorporated.
Return the speed to medium, and let the dough need for an additional 10-15 minutes, or until the window pane test works. Transfer the dough to a lightly-oiled bowl and cover with plastic wrap. Let the dough rise until it is about double in size. For me this takes about 30 minutes, but pay attention to the size most of all.
Once risen, scrape the dough onto a floured surface. It will still be fairly sticky at this point, so be generous with the flour. Roll out to about 1/2 inch thick. Cut the dough into your desired shapes.
Line two large sheet pans with parchment paper, and a third sheet pan with several layers of paper towels. Place your cut doughnuts onto the parchment lined sheets, and cover with plastic. Let them rise for about 10-15 minutes, or until they spring back when touched.
Meanwhile, in a deep pot, heat about 4 cups of canola or vegetable oil to 350-375 degrees.
Fry the doughnuts for about 1 minute on each side, or until they become a deep golden brown. Remove from the oil and place onto the paper towel lined sheet pan to drain. Repeat with the remaining doughnuts- you only want to fry about 3 at a time. Coat in any icing or sugar mixture you prefer!