Custard-Filled Donuts

In a medium saucepan, heat the milk over medium heat until it starts to simmer.
In a separate bowl, whisk together the granulated sugar, cornstarch, and egg yolks until well combined.
Slowly pour the hot milk into the egg mixture, whisking constantly to temper the eggs.
Pour the mixture back into the saucepan and cook over medium heat, stirring constantly, until it thickens and comes to a boil.
Remove from heat and stir in the butter and vanilla extract. Transfer the custard to a bowl, cover with plastic wrap (pressing it directly onto the surface to prevent a skin from forming), and let it cool completely in the refrigerator.
Shape and Fry the Donuts:
Punch down the risen dough and roll it out to about 1/2 inch thickness on a floured surface.
Cut out rounds using a donut cutter or a round cookie cutter.
Place the donut rounds on a baking sheet lined with parchment paper, cover with a clean cloth, and let them rise for about 30 minutes.
Heat the vegetable oil in a large pot or deep fryer to 350°F (175°C).
Fry the donuts in batches for about 1-2 minutes per side, or until golden brown. Remove with a slotted spoon and drain on paper towels.
Fill the Donuts:
Once the donuts are cool enough to handle, fill a piping bag fitted with a small round tip with the cooled custard filling.
Poke a hole in the side of each donut and pipe the custard inside until filled.