Prepare the Dough:

1. Activate the Yeast: In a small bowl, combine warm milk (about 110°F) and a teaspoon of sugar. Sprinkle the yeast on top and let it sit for 5-10 minutes until it becomes frothy.
2. Mix the Dough: In a large bowl, combine flour, sugar, and salt. Add the frothy yeast mixture, eggs, and butter. Mix until a dough forms.
3. Knead the Dough: Knead the dough on a floured surface for 8-10 minutes until smooth and elastic. Alternatively, use a stand mixer with a dough hook.
4. First Rise: Place the dough in a greased bowl, cover it with a damp cloth, and let it rise in a warm place for 1-1.5 hours or until doubled in size.

Make the Chocolate Custard:

5. Heat the Milk: In a saucepan over medium heat, warm the milk until it just begins to simmer.
6. Whisk the Eggs and Sugar: In a separate bowl, whisk together the egg yolks, sugar, cornstarch, and salt until smooth.
7. Temper the Eggs: Gradually add a ladle of the warm milk to the egg mixture, whisking constantly to prevent curdling.
8. Cook the Custard: Pour the tempered egg mixture back into the saucepan. Cook over medium heat, stirring constantly, until the custard thickens (about 5-7 minutes).
9. Add the Chocolate and Vanilla: Remove from heat, stir in the chocolate and vanilla until smooth. Cover with plastic wrap (pressing it directly onto the surface) and refrigerate until chilled.

Shape and Fry the Donuts:

10. Roll Out the Dough: After the first rise, roll the dough to about ½-inch thickness on a floured surface.
11. Cut the Donuts: Use a round cutter (about 3 inches) to cut out circles. Place them on a parchment-lined baking sheet.
12. Second Rise: Cover the donuts with a cloth and let them rise for another 30-45 minutes until puffed.
13. Heat the Oil: Heat 2-3 inches of oil in a deep pot to 350°F (175°C).
14. Fry the Donuts: Fry 2-3 donuts at a time, cooking for 1-2 minutes on each side until golden brown. Transfer to a wire rack to cool.

Fill the Donuts:

15. Fill with Custard: Once the donuts are cool, poke a hole in each donut using a skewer or piping tip. Fill a piping bag with the chocolate custard and pipe it into each donut.
16. Coat the Donuts: Roll the filled donuts in granulated or powdered sugar, if desired.