Filled Sourdough Donuts (Bomboloni or Krapfen)

Steps

1. 1Prepare tangzhong mixture: Whisk together milk and flour in a small saucepan and set over medium heat. Stir constantly with a rubber spatula until it thickens to the consistency of mashed potatoes. Spread the mixture flat on a dinner plate and cover the surface with plastic wrap. Set aside (or refrigerate) until completely cool.
2. 2Mix & knead dough: Into the bowl of a mixer fitted with a dough hook, add cooled tangzhong, ripe starter, eggs, sugar, and flour. Mix on low until combined. Cover dough and allow to rest for 30 minutes. Add the salt and knead for 10 minutes. The dough should look fairly strong at this point. Add half the butter cubes and continue to mix on low until the butter is fully incorporated. Add the remaining butter cubes and continue to knead until the butter is no longer visible. Assess the dough. lf dough is very smooth, shiny, pulls itself away from the bowl to hold to the dough hook, and can be pulled to a dramatic "windowpane" with damp fingers, it's done. If it's not yet showing these signs, knead for a couple more minutes and check again.
3. 3Bulk ferment and refrigerate overnight: Turn out the dough onto a clean countertop. Use a damp bench scraper to form the dough into a smooth ball by scooting it against the counter in a circular motion. Place dough in a large bowl, cover, and allow to rise in a warm place until it increases in volume by 1/3. This takes 3-4 hours at 80°F (26°C) (see notes below). Place in the refrigerator overnight.
4. 4Shape: Prepare two half sheet pans with six 5-inch squares of parchment paper each, and dust the parchment squares with flour.Divide half of the cold dough into 12 equal pieces (weigh for accuracy, if desired). The remaining dough can be used for more donuts, another baking project, or used the following day. Lightly flouring your hands as needed, press each piece of dough with your hands to degas, then form into a tight ball by scooting in a circular motion with your hand against the countertop. Arrange the balls on the squares of floured parchment paper.
5. 5Proof: Cover trays with plastic wrap and set in a warm place to proof until they have doubled in size and pass the "poke" test. (This will take about 3.5 hours at 78°F, longer if your kitchen is cooler.)
6. 6Fry: Meanwhile, line a cooling rack with a double layer of paper towels. Fill a bowl or small plate with sugar for rolling. Fill a heavy-bottomed pot with oil and set over medium heat until oil reaches 360°F (use a candy thermometer). Keep oil between 350°F and 370°F while frying. Fry up to 4 donuts at a time, flipping after the first sides are deeply golden to fry the other sides (about 2 minutes per side). (The parchment papers will allow you to easily transfer the donuts to the frying oil, releasing as you drop the donuts in top-side down.) Remove donuts with a slotted spoon, allowing excess oil to drip off into the pot, then place on the paper towels to drain.
7. 7Roll in sugar and fill: As soon as donuts are cool enough to handle, roll them in the sugar. Prepare a piping bag with a large round tip (or just cut off tip) and fill with desired filling. Poke a hole into the side of the first donut with a pairing knife until you reach the center of the donut. Give the knife a twist to make a space for the piping bag. Pipe in desired amount of filling (aim for 1-2 tablespoons), and repeat for remaining donuts. Serve donuts the same day. (The frying oil can be cooled, strained, and stored for next time.)