Mix Dough Ingredients:
In the bowl of a stand mixer fitted with the dough hook, combine the bread flour, sugar, yeast, and salt. Add the milk, egg, and tangzhong. Knead on low speed for 2 minutes until a dough forms.Knead with Butter:
With the mixer running on low, add the softened butter a tablespoon at a time, letting it fully incorporate before adding more. Once all the butter is added, increase the speed to 2 on the KitchenAid and knead for 12 minutes, or until the dough is smooth, elastic, and passes the windowpane test.Shape and Rest Dough:
Shape the dough into a ball and place it in a lightly greased bowl. Cover and rest for 20–30 minutes to relax the dough.Cut Parchment Squares:
While the dough is resting, cut out 9 squares of parchment paper.Cut Donut Rounds:
Turn the dough onto a floured surface, lightly flour the top, and roll to 3/4-inch thickness. Use a large ~3-inch round cutter to cut donuts. Gently re-roll the scraps once, then rest the dough for 5–10 minutes before cutting again.Proof Donuts:
Place each donut on a square of parchment paper and arrange them on a baking tray. Cover the top loosely with plastic wrap and let rise until puffy and doubled, about 45–65 minutes.Heat Oil:
When the donuts are about halfway through their final proof, start heating 3–4 inches of oil in a heavy pot over medium-low heat to 350°F (175°C). Line a baking sheet with a wire rack. Keep an eye on the temperature with a candy thermometer. If the oil reaches 350°F (175°C) before the donuts are ready, lower the heat slightly to hold it steady until they finish proofing.Fry Donuts:
You’ll know the donuts are ready when you gently touch the side with your finger and the dough feels slightly airy. Use a metal slotted spoon to lower 1-2 donuts into the oil with the parchment paper on. After 5-10 seconds, use tongs to remove the paper from the oil. Each donut will fry for 2 1/2 minutes total, flipping once halfway through. Use a timer so you don’t overcook them! Once fried, transfer them to the prepared wire rack.