-
To make the tangzhong: Whisk the tangzhong ingredients in a small saucepan until no lumps remain. Place the saucepan over medium heat and cook, stirring regularly, until the mixture is thickened and paste-like and a dragged spoon or spatula leaves a trail in the bottom of the pan, 2 to 3 minutes. Remove the saucepan from the heat and scrape the tangzhong into the bowl of a stand mixer fitted with the dough hook.
-
To make the dough: Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the flour, milk, starter, sugar, egg, yeast, and salt to the tangzhong in the mixer bowl.
-
Mix on low speed to bring the dough together, about 1 minute, then pause to scrape the bowl. Next, mix on medium speed until the dough is smooth, sticky, and shiny, 2 to 3 minutes.
-
With the mixer running, add the butter 1 tablespoon at a time, mixing until fully incorporated after each addition. Once all the butter has been added, transfer the dough to a lightly greased bowl.
-
Cover the bowl and let the dough rise at room temperature for 1 hour, or until slightly puffy but not doubled in volume.
-
Transfer the bowl to the refrigerator for at least 1 hour and up to 24 hours. While the dough is chilling, gather parchment squares or cut ten 4" squares of parchment.
-
To shape the doughnuts: Turn the chilled dough out onto a lightly floured work surface. Use a rolling pin or your hands to gently flatten the dough to an even 1/2" thickness.
-
Use a floured 3 1/2" round cutter to cut out doughnuts, then use a floured 1 1/2" round cutter to cut a hole out of the center of each. Place the doughnuts on separate parchment squares (the holes can be grouped together on one square) and transfer on the parchment to a baking sheet. Cover with lightly greased plastic wrap.
-
Gather the dough scraps and gently reroll them to an even 1/2" thickness. Cut out more doughnuts and doughnut holes until you have 9 doughnuts and 9 doughnut holes.
-
Let the doughnuts rise at room temperature for 15 to 30 minutes, or until they are slightly puffy and hold an indentation when gently pressed with a floured fingertip.
-
To fry the doughnuts: Toward the end of the rising time, heat the oil to 325°F in a large, heavy pot. Line a baking sheet with paper towels and set a wire rack on top.
-
Fry 3 doughnuts at a time, using the parchment squares to carefully invert each doughnut into the hot oil (parchment side up); the parchment should quickly release – use tongs to dispose of it. (If it doesn’t release, don’t worry: It will when you flip the doughnuts.) Cook the doughnuts until they’re brown on the bottom, 1 1/2 to 2 minutes. Using a slotted spoon, turn the doughnuts over and cook until brown on the other side, another 1 1/2 to 2 minutes. Adjust the heat as necessary to keep the oil temperature between 325°F and 350°F.
-
Transfer the doughnuts to the wire rack to cool completely. Repeat until all the doughnuts are fried.
-
Fry the doughnut holes in a single batch, gently stirring and turning them constantly with the slotted spoon, until brown all over, 3 to 4 minutes. Transfer to the wire rack and let cool completely.
-
To make the glaze: While the doughnuts are cooling, whisk the glaze ingredients in a large bowl until smooth.
-
Dip the cooled doughnuts and doughnut holes into the glaze, fully coating one side, then return them to the rack to let the glaze set, about 1 minute.
-
Storage information: Sourdough doughnuts are best enjoyed the day they are made. Leftover doughnuts can be stored in a paper bag at room temperature for up to 1 day. They taste best when reheated in the microwave for 10 seconds.