Combine the warm water, milk, sugar, and yeast in a large bowl. Whisk to dissolve the yeast. Allow the yeast to proof for about 5 minutes or until it starts to foam.

Add the remaining ingredients. Knead on low using a dough hook on your mixer or by hand. Knead for about 5 minutes or until the dough is nice and smooth.

Place the dough into a greased bowl, cover with plastic wrap and let rise in a warm draft-free place until almost doubled in size, about an hour.

Punch the dough down and roll out on a lightly floured surface to 1/2-inch thick. Use a 3-inch cutter to cut the donuts. Place the donuts on a floured parchment-lined baking sheet, cover with plastic wrap, and let proof for 15 minutes. Re-roll the remaining dough, cut out the remaining donuts and add them to the baking sheet.

Heat 2 inches of oil in a heavy pot (Dutch oven) to 350-360°F. Drop 2-3 donuts at a time and allow to become golden before turning to the other side, about 2 minutes per side.

Remove from the oil and place on paper towel-covered wire rack. Let cool before filling.

For the pastry cream, add the sugar, cornstarch, and milk to a large saucepan and mix until smooth. Cook, stirring continuously, over medium heat until mixture begins to thicken and boil. Reduce heat to medium and simmer for 1 minute. Remove from heat.

Add a little bit of mixture to the egg yolks and whisk together, then add egg mixture to milk mixture. Place the saucepan back on the heat and bring to a light boil. Allow to boil for 2 minutes, stirring continuously.

Remove from heat, add butter and vanilla. Stir until smooth. Pour into a medium bowl. Cover with plastic wrap and press it directly on surface. Refrigerate until cold.

For the chocolate glaze, heat the cream in a microwave or over the medium heat until just below boiling point. Pour the heated cream over the chocolate chips and let sit for 1 minute. Whisk it all together until the chocolate it fully melted.

Fill a piping bag with a round tip with the cream. Using a small knife, cut a hole in the side of each donut about 3/4 of the way into the donut and pipe a generous amount of vanilla cream into the center of the donut.

Swirl the top of the donut in the glaze and allow to sit for about 10 minutes before serving.