In the bowl of a stand mixer fitted with the dough hook attachment, combine the active sourdough starter, warm milk, flour, eggs, salt, granulated sugar, and vanilla.

Mix on low speed until the ingredients come together into a shaggy dough.

With the mixer running on low speed, add the softened butter one tablespoon at a time, allowing each addition to fully incorporate before adding the next.

Transfer the dough to a large lightly greased bowl and cover tightly with plastic wrap.

Allow the dough to bulk ferment in a warm place for 6–10 hours, or until it has risen approximately 50–60%. Because this is a rich brioche dough containing butter and eggs, it will ferment more slowly than a lean sourdough dough.

Once the bulk fermentation is complete, cover the bowl tightly and refrigerate overnight. Chilling the dough makes it much easier to roll, cut, and shape. Alternatively, you can let the dough bulk ferment overnight (if your kitchen allows it) and shape right away the next day.

The next day, prepare the chocolate custard while the dough is still chilled.

In a medium mixing bowl, whisk together the egg yolks and cornstarch until completely smooth. Set aside.

In a medium saucepan over medium heat, combine the whole milk, granulated sugar, and salt. Heat until the mixture is steaming and just beginning to simmer around the edges, whisking occasionally.

Slowly ladle about 240 g (1 cup) of the hot milk mixture into the egg yolk mixture while whisking constantly to temper the eggs.

Gradually pour the tempered mixture back into the saucepan while whisking continuously.

Continue cooking over medium heat, whisking constantly, for 3–5 minutes, until the custard becomes thick and smooth.

Remove the saucepan from the heat and immediately stir in the milk chocolate melting wafers. Whisk until the chocolate has completely melted and the custard is silky smooth. Stir in the vanilla extract.

Transfer the chocolate custard to a clean bowl and press plastic wrap directly against the surface to prevent a skin from forming.

Refrigerate while you shape and proof the donuts. The custard will continue to chill and thicken, making it perfect for piping later.

Remove the chilled dough from the refrigerator and transfer it to a lightly floured work surface.

Roll the dough to approximately ½-inch thick.

Using a round cookie cutter or the rim of a drinking glass, cut out donut rounds. Gather the scraps, reroll, and continue cutting until all of the dough has been used.

Cover loosely with plastic wrap or a clean kitchen towel.

Allow the donuts to rise in a warm place for 2–4 hours, or until they are noticeably puffy, light, and nearly doubled in size. The exact proofing time will vary depending on the temperature of your kitchen.

Working in batches, carefully lower the donuts into the hot oil. If desired, use the parchment squares to transfer the donuts into the oil, then remove the parchment with tongs once the donut releases.

Fry for 1–2 minutes per side, or until both sides are deep golden brown.

While still warm, roll each donut generously in granulated sugar until evenly coated.

Allow the donuts to cool completely before filling.

Transfer the chilled chocolate custard to a piping bag fitted with a piping tip.

Using a butter knife or skewer, create a small opening in the side of each cooled donut.

Insert the piping tip into the opening and pipe the chocolate custard into each donut until filled. You'll feel the donut become slightly heavier as the custard fills the center.

Serve immediately, or refrigerate until ready to enjoy.