Step 1In the bowl of a stand mixer fitted with the dough hook, add the warmed milk, then the yeast and whisk by hand to combine. Beat in the eggs, then add the flours, ¼ cup plus 1 teaspoon sugar and 1 ½ teaspoons salt. Mix on low speed until the dough comes together, 1 to 2 minutes.
Step 2Increase the mixer speed to medium-high for about 6 minutes to work the dough. Pause about every minute or so to push the dough back down into the bowl and off the hook.
Step 3Reduce the mixer speed to low and slowly add the butter, a little at a time, over the course of 2 minutes. Pause halfway though to scrape down the bowl and the hook. When the butter begins to blend in, increase the mixer speed to medium-high to fully incorporate the butter and bring the dough back together, 4 to 6 minutes longer.
Step 4Transfer the dough to a greased bowl, cover with plastic wrap, and refrigerate overnight.
Step 5Remove the dough from the refrigerator and dump it onto a clean, well-floured surface. Sprinkle more flour over the dough. Flatten the dough by hand or by rolling it to a thickness of 1 ½ to 1 ¾ inches.
Step 6Using a 3-inch-round cutter, cut 10 rounds and transfer them to a greased sheet pan, spacing them 1 ½ inches apart. Using a 1/2-inch round cutter, cut a small hole in each round and place them on the pan as well. Use the leftover dough to cut more holes and place them on the sheet pan too.
Step 7Loosely cover the dough with plastic wrap and set aside to rise at room temperature, or refrigerate overnight and allow to rise in the morning, until doubled in size, 1 to 2 hours.
Step 8Meanwhile, in a bowl, stir together the remaining 2 cups sugar, the cinnamon and remaining ¼ teaspoon salt. Set aside.
Step 9Heat 3 inches of oil to 375 degrees in a Dutch oven or countertop fryer. Begin frying 1 test doughnut for about 2 minutes on each side, then remove it from the oven and check to see if it’s cooked through. If it’s browned on the outside but raw inside, the dough is underproofed and hasn’t risen enough. Stop frying, re-cover the doughnuts and set them aside to continue to rise and try again.
Step 10When the dough is ready, place each doughnut onto a long-handled skimmer, and carefully lower it into the oil. Working in batches, fry 3 to 4 doughnuts at a time until they are a nice golden brown, about 2 minutes per side, then fry all the holes at once until golden.
Step 11Immediately toss the doughnuts in the cinnamon-sugar and serve hot.