In the bowl of stand mixer, combine milk, yeast, and sugar. Allow mixture to sit for 5 minutes. Whisk in egg until incorporated.
In a large bowl, whisk together flour and salt. Attach dough hook attachment to stand mixer. With the mixer running on low, add 1/3 of flour mixture. Once dough starts to come together, add another 1/3 of flour mixture. Add softened butter one tablespoon at a time, mixing well after each addition. At this point the dough will be sticky. Scrape down bowl as needed to ensure thorough mixing. Add remaining 1/3 of flour mixture.
Once dough starts to form, increase mixer speed to medium. Knead for 3-5 minutes until dough is smooth and soft. Dough may still stick to the sides of the bowl, but it should be easy to scrape using a spatula or dough scraper. (Do not add additional flour)
Transfer dough to a large greased bowl. Cover with plastic wrap or kitchen towel and allow dough to rest for 30-45 minutes at room temperature until dough has doubled in volume.
Gently deflate risen dough. Working on a lightly floured surface divide dough into 12 equal parts. Loosely cover dough with kitchen towel. Working with one piece at a time, stretch and roll dough into a 14-inch stripe.
To form twist, place hands on ends of rope. Roll one hand upward while simultaneously rolling other hand downward. This will cause the dough to twist. Pick up the twisted dough in the center of the rope. The tension in the twisted dough will naturally braid together. Pinch end seam together and place on a parchment lined baking sheet. Loosely cover with kitchen towel and repeat with remaining dough.*
Allow dough to rest for 8-10 minutes while oil heats up. Attach kitchen thermometer to a heavy bottomed sauce pot (such as Dutch oven or cast iron pan). Fill pot with 2 to 3 inches of oil. Set over medium heat until oil reaches 350°F.
Carefully place two or three doughnut twists into hot oil. Do not overcrowd the pot. Cook for 95 seconds on each side, until golden brown. Transfer cooked doughnuts on a paper towel lined baking sheet. NOTE: Oil temperature will change as you add raw doughnuts into the pot. Monitor oil temperature using thermometer to make sure the doughnuts do not burn or undercook.
Cinnamon Sugar: combine sugar and cinnamon in a shallow dish. Set aside until ready to coat cooked doughnuts.
Toss warm doughnuts in cinnamon sugar coating. Enjoy immediately.