Prepare Tangzhong Mixture: In a small saucepan, whisk together the milk and flour over medium heat. Stir constantly with a rubber spatula until the mixture thickens to a mashed potato-like consistency. Spread this mixture flat on a dinner plate and cover with plastic wrap, ensuring the wrap touches the surface. Set aside or refrigerate until fully cooled.

Mix & Knead Dough: In the bowl of a mixer fitted with a dough hook, combine the cooled tangzhong, ripe starter, eggs, sugar, and flour. Mix on low until just combined. Cover the dough and let it rest for 30 minutes. After resting, add the salt and knead for 10 minutes. The dough should start to look strong. Incorporate half of the butter cubes and mix on low until fully absorbed. Add the remaining butter and continue kneading until the dough is smooth, shiny, and passes the "windowpane" test.

Bulk Ferment & Refrigerate Overnight: Turn the dough onto a clean surface. Using a damp bench scraper, shape it into a smooth ball by scooting it in a circular motion against the counter. Place the dough in a large bowl, cover it, and let it rise in a warm place until it increases in volume by one-third (approximately 3-4 hours at 80°F/26°C). Then, refrigerate overnight.

Shape Dough: Prepare two half-sheet pans with six 5-inch squares of parchment paper each, dusting the parchment with flour. Divide the cold dough into 12 equal pieces (weigh for accuracy if desired). The remaining dough can be used for more donuts or other baking projects. Lightly flour your hands as needed, press each dough piece to degas, and then shape into tight balls by scooting them in a circular motion against the counter. Place each ball on a floured parchment square.

Proof: Cover the trays with plastic wrap and let them proof in a warm place until doubled in size and they pass the "poke" test (approximately 3.5 hours at 78°F, longer if your kitchen is cooler).

Fry: Line a cooling rack with a double layer of paper towels and prepare a bowl of sugar for rolling the donuts. Fill a heavy-bottomed pot with oil and heat over medium until it reaches 360°F (using a candy thermometer). Keep the oil temperature between 350°F and 370°F while frying. Fry up to four donuts at a time, flipping them once the first side is deeply golden (about 2 minutes per side). Use the parchment papers to transfer the donuts into the oil, releasing them top-side down. Remove the donuts with a slotted spoon, letting excess oil drip back into the pot, and then place them on the paper towels to drain.

Roll in Sugar & Fill: Once the donuts are cool enough to handle, roll them in the sugar. Fill a piping bag fitted with a large round tip (or simply cut off the tip) with your desired filling. Poke a hole in the side of each donut with a paring knife, twist to create space, and pipe in 1-2 tablespoons of filling. Repeat for the remaining donuts. Serve the donuts the same day for optimal freshness. (The frying oil can be cooled, strained, and stored for future use.)