In a large mixer bowl, add All-Purpose Flour (3 1/3 cup), Granulated Sugar (1/4 cup), Active Dry Yeast (2 1/4 teaspoon), and Salt (1 teaspoon). Mix and set aside.
In a separate jar, mix butter and milk and heat in the microwave to get Butter (4 tablespoon) and Milk (1 cup) Stir in Vanilla Extract (1 teaspoon).
In the bowl with the dry ingredients, add in 1 whole Egg (1), then add wet ingredients.
Use a mixer on low until it becomes clumpy, then increase the speed and continue to mix until the dough is nice and smooth.
Grease the same mixing bowl, then place the dough back inside and rub some of the grease around the dough. Cover with plastic wrap and set a kitchen towel on top. Let it rise for 1 1/2 - 3 hours until it has doubled in size.
To make the custard, add Milk (1 cup), Granulated Sugar (1/4 cup), yolks from the remaining Egg (2), Salt (1 pinch), Corn Starch (2 tablespoon), Vanilla Extract (1 teaspoon) to a large saucepan and mix.
Put the saucepan on low heat on the stove, stirring constantly. Cook until the mixture thickens.
Take the thickened mixture and sift through a strainer into a bowl.
Cover the custard with plastic wrap, making sure that the plastic touches the top of the mixture to prevent a skin from forming. Leave in fridge until thick.
Once the dough has risen, place it on a floured surface and push it down with your hands, then use a rolling pin to roll it out to 1/2 an inch thick.
Use a cup or other round object to punch out donut circles from the dough and place each circle on a floured baking sheet. Cover with a kitchen towel and let rise for 30 - 45 minutes.
In a saucepan, heat oil to about 350 degrees F (180 degrees C). Dip risen dough circles into oil for about 1 minute until golden brown.
Allow donuts to cool completely, then take a knife and make a hole in the donut and through the hole pipe the custard into the donut. Set aside.
To make the chocolate glaze, add Powdered Confectioners Sugar (1 cup), Unsweetened Cocoa Powder (3 teaspoon), Vanilla Extract (1 teaspoon), Milk (to taste) to a bowl and mix together.
Dip the donuts in the glaze, let dry, and enjoy!