Donuts filled with vanilla pudding!

1. Vanilla pudding (to be prepared in advance):
Heat the milk with the vanilla.
Whisk the egg yolks with the sugar, then add the cornstarch.
Pour in the hot milk, stirring constantly, and return the mixture to the stove over medium heat.
Stir until thickened, then add the butter.
Cover with plastic wrap and let cool completely.
2. The donut dough:
Dissolve the yeast in the warm milk and let it rest for 10 minutes.
In a large bowl, combine the flour, sugar, salt, eggs, and milk/yeast.
Knead for 10 minutes (by hand or with a stand mixer), then add the softened butter.
Continue kneading until a smooth and elastic dough forms.
Cover and let rise for 1.5 hours, or until doubled in size.
3. Shaping and baking:
Roll out the dough to 1.5 cm thick and cut into circles.
Let rise for another 30-45 minutes on a floured baking sheet.
Fry the donuts over medium heat until golden brown on both sides.
Drain on paper towels and then roll in sugar.
4. Filling:
Using a piping bag, fill each donut with vanilla pudding.