In a stand mixer bowl, combine warm milk (110°F), yeast, and 3 tablespoons sugar. Let sit for 5 minutes until the mixture becomes foamy and bubbles form on the surface. This means the yeast is activated.
Whisk the egg into the yeast mixture until fully incorporated with no streaky egg whites visible.
In a separate bowl, whisk together flour and salt to ensure even distribution.
Attach the dough hook to your stand mixer. With mixer on low speed, add one-third of the flour mixture. Once it starts clumping together, add the second third of flour.
Add the softened butter one tablespoon at a time, waiting for each piece to fully incorporate before adding the next.
Add the final third of flour. Increase mixer speed to medium and knead for 3-5 minutes until the dough pulls away from the bowl sides and develops a smooth, satiny appearance.
Transfer dough to a greased bowl and lightly coat the surface with oil. Cover with plastic wrap or a clean kitchen towel. Let rise at room temperature for 35-45 minutes, or until doubled in size.
Gently press down the risen dough to release gases. Divide into 12 equal portions on a lightly floured surface. Cover portions with a kitchen towel while working.
Take one portion and roll it between your palms and the work surface into a 14-inch rope with even thickness throughout.
Hold both ends of the rope. Roll your right hand forward and left hand backward simultaneously to create tension. Lift the rope by both ends and bring them together - the dough will naturally spiral. Give it 1-2 more deliberate twists for a tighter braid. Pinch the ends together firmly.
Place twisted doughnuts on a parchment-lined baking sheet and cover loosely. Let rest for 8-10 minutes.
Attach a candy thermometer to a heavy-bottomed pot or Dutch oven. Fill with 2-3 inches of vegetable oil. Heat over medium heat to exactly 350°F.
Carefully slide 2-3 twists into the hot oil without crowding. Fry for 95 seconds, then flip gently with a spider strainer or slotted spoon. Fry for another 95 seconds until golden brown.
Transfer fried doughnuts to a paper towel-lined baking sheet to drain. Monitor and adjust heat between batches to maintain 350°F oil temperature.
Mix cinnamon and ½ cup sugar in a shallow dish. While doughnuts are still warm (but not scorching hot), toss each one in the cinnamon sugar mixture until completely coated.