In a bowl, combine 1 cup lukewarm milk, 3 tablespoons sugar, and the yeast.
Let it sit for 10-15 minutes, until foamy.
Add 2 eggs, salt, and flour. Mix on medium speed with the dough hook attachment for about 5 minutes until the dough starts to come off the side.
Start adding butter, one tablespoon at a time, until all 6 tablespoons are added in, about 10-15 minutes. Before adding each portion of butter, make sure the previous one is fully incorporated into the dough.
Place the dough in a greased bowl, cover it with a towel, and let it sit in a warm place until it's double the size, about an hour.
Meanwhile, slice the vanilla bean in half lengthwise and scrape the seeds.
Add 2 cups milk, the vanilla seeds, and the bean to a saucepan and bring to a simmer over low heat. Let it simmer for 5 minutes.
In a bowl, mix the egg yolks, ½ cup sugar, cornstarch, and salt until smooth.
Pour some of the hot milk over the egg yolk mixture to temper the eggs, whisking constantly.
Pour the tempered egg mixture back into the rest of the milk and cook on medium heat, stirring constantly, until thick, about a minute.
Take the mixture off the heat and mix in 3 tablespoons of butter until fully incorporated.
Run the cream through a sieve.
Cover the cream with plastic wrap, touching the surface of the cream to prevent it from forming a skin. Transfer to the fridge to cool.
On a floured surface roll the dough to ¾-inch thickness.
Cut out the donuts using a glass or a donut cutter about 3 inches in diameter.
Place the cut-out donuts on parchment paper and cut out the paper around each one for easier transfer to the hot oil.
Cover the donuts and let them rest for 30-60 minutes.
Heat the oil in a deep large saucepan until it reaches 350 F.
Place no more than 3 donuts at a time in the oil.
Cook for about 2 minutes on each side, or until golden brown. Repeat with the rest of the donuts.
Let the donuts cool completely.
Make the chocolate sauce by melting the chocolate over a double boiler.
Mix in ¼ cup milk.
Cut off the end of a pastry bag or a zip-top bag and place a round piping tip in it.
Fill the bag with the cooled pastry cream.
Make a hole in each donut, using a knife. Make sure not to cut through the other end.
Fill each donut with the cream.
Dip the tops of each donut in the chocolate sauce.