Start by combining the warm water, lukewarm milk, granulated sugar, and dry yeast in a large mixing bowl. Whisk everything together until the yeast dissolves. Allow the mixture to sit for about 5 minutes until it becomes foamy, indicating that the yeast is active.

Next, incorporate the remaining ingredients into the bowl. Using a dough hook on your mixer or kneading by hand, work the dough on a low setting for about 5 minutes until it becomes smooth and elastic.

Once the dough is ready, place it in a greased bowl, cover it with plastic wrap, and let it rise in a warm, draft-free area until it has nearly doubled in size, which should take around one hour.

After the dough has risen, punch it down and roll it out on a lightly floured surface to a thickness of 1/2 inch. Use a 3-inch cutter to cut out the donuts. Transfer the donuts to a parchment-lined baking sheet dusted with flour, cover them with plastic wrap, and allow them to proof for 15 minutes. Gather the scraps of dough, re-roll, and cut additional donuts, placing them on the same sheet.

In a heavy pot, such as a Dutch oven, heat 2 inches of oil to 350-360°F. Fry 2-3 donuts at a time, allowing each side to turn golden brown, about 2 minutes per side. Once fried, remove the donuts from the oil and place them on a paper towel-lined wire rack to cool before filling.

To prepare the custard, combine the granulated sugar, cornstarch, and milk in a large saucepan, stirring until smooth. Cook the mixture over medium heat, stirring constantly, until it thickens and begins to boil. Lower the heat to medium and simmer for 1 minute before removing from the heat.

Temper the egg yolks by adding a small amount of the hot mixture to them, whisking quickly. Then, add the egg mixture back to the saucepan with the milk mixture. Return the pan to the heat and bring it to a gentle boil, allowing it to cook for an additional 2 minutes while stirring continuously.

Remove the saucepan from the heat and stir in the unsalted butter and vanilla extract until smooth. Pour the custard into a medium bowl, cover it with plastic wrap pressed directly onto the surface to prevent a skin from forming, and refrigerate until completely chilled.

For the chocolate glaze, heat the heavy cream until it’s just below the boiling point. Pour the hot cream over the semisweet chocolate chips and let it sit for 1 minute. Then, whisk the mixture until the chocolate is fully melted and smooth.

To assemble, fill a piping bag fitted with a round tip with the chilled custard. Using a small knife, make a hole in the side of each donut, extending about 3/4 of the way into the center. Pipe a generous amount of custard into each donut.

Finish by dipping the top of each donut into the chocolate glaze, swirling slightly to ensure an even coating. Allow the donuts to sit for about 10 minutes for the glaze to set before serving.