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 milk, sugar, and salt. Whisk and heat until the mixture is steaming but not boiling.

Scoop out about 1 cup (240 g) of the hot milk mixture and slowly drizzle it into the egg mixture while whisking constantly. This tempers the eggs and prevents scrambling.

Gradually pour the tempered egg mixture back into the saucepan with the remaining milk, whisking continuously.

Cook over medium heat, whisking constantly, for 3–5 minutes or until thickened.

Remove from heat and stir in the vanilla extract.

Transfer custard to a clean bowl. Press plastic wrap directly onto the surface to prevent a skin from forming. Refrigerate until completely chilled (at least 2 hours).

In a small bowl, combine the warm milk, yeast, and sugar. Let sit for 5–10 minutes until foamy.

In the bowl of a stand mixer fitted with a dough hook, add the yeast mixture, egg, salt, sourdough discard, and flour.

Mix on medium speed until combined, scraping down the bowl as needed.

With the mixer running, add the softened butter one tablespoon at a time, allowing each addition to incorporate before adding more.

Knead on medium-high speed for 8–10 minutes, until the dough is soft, smooth, and pulling away from the sides of the bowl.

Transfer the dough to a lightly oiled bowl.

Cover and let rise in a warm place for about 2 hours, or until doubled in size.

Punch down the dough and roll it out on a lightly floured surface to about 1/2-inch thickness.

Heat 2–3 inches of neutral oil in a heavy-bottomed pot to 340–360°F.

Carefully lower donuts into the hot oil (working in batches).

Fry 1–2 minutes per side, until golden brown.

Transfer to a wire rack with parchment underneath.

While still warm, roll each donut in granulated sugar. Return to rack to cool.

Transfer chilled custard to a piping bag fitted with a round tip.

Once donuts are fully cooled, use a butter knife to create a small hole in the side or center.

Pipe custard into each donut until filled.