In a medium saucepan, whisk together egg yolks, cornstarch, whole milk, granulated sugar, salt, and vanilla extract or paste. Cook over medium heat, stirring constantly, until the mixture thickens. Remove from heat and let it cool. Cover and refrigerate until ready to use.

In a small bowl, combine warm milk, yeast, and 1/4 cup granulated sugar. Let it sit until frothy, about 5 minutes.

In a large bowl, mix together the egg, salt, sourdough discard, and yeast mixture.

Gradually add the flour and knead until incorporated. Fold in the butter until the dough is smooth.

Cover the dough with a damp cloth and let it rise in a warm place until doubled in size, about 1 to 1.5 hours.

Punch down the dough and roll it out on a lightly floured surface.

Cut out donut shapes and place them on a parchment-lined baking sheet. Cover with a cloth and let them rise again until puffy, about 30 to 45 minutes.

Heat the oil in a deep pot to 350°F (175°C). Carefully fry the donuts in batches until golden brown, about 2 minutes per side. Drain on paper towels.

Once the donuts are cool, fill them with the chilled custard using a pastry bag or injector. Roll in granulated sugar before serving.