In a big bowl, mix together the warm water, milk, sugar, and yeast. Use a whisk to dissolve the yeast fully. Let the yeast sit and activate for approximately 5 minutes, or until it begins to form bubbles or foam.

Include the rest of the components and mix them together. Use a dough hook on your mixer or do it manually by hand on a low setting. Continue kneading for approximately 5 minutes or until the dough becomes smooth and appealing.

Put the dough in a bowl that has been coated with grease, cover it with plastic wrap, and allow it to increase in size by almost double in a warm area without any air movement. This process should take approximately one hour.

Deflate the dough and flatten it on a surface sprinkled with flour until it is 1/2-inch thick. Cut out the donuts using a 3-inch cutter. Arrange the donuts on a baking sheet lined with parchment paper and flour. Cover with plastic wrap and allow them to rise for 15 minutes. Flatten the remaining dough again, cut out the remaining donuts, and place them on the same baking sheet.

In a large and sturdy pot, such as a Dutch oven, heat about 2 inches of oil until it reaches a temperature between 350-360°F. Put in 2-3 donuts at once and let them cook until they turn a golden color on one side. This usually takes around 2 minutes per side.

Take out of the oil and put on a wire rack that is covered with paper towel. Wait until it cools down before adding any filling.

To make the pastry cream, combine the sugar, cornstarch, and milk in a saucepan, and stir until well-blended. Heat the mixture over medium heat, stirring consistently, until it starts to thicken and come to a boil. Lower the heat to medium and let it simmer for one minute. Take the saucepan off the heat.

Combine a small amount of mixture with the egg yolks and mix thoroughly. Then, add the egg mixture to the milk mixture. Return the saucepan to the stove and heat until it gently boils. Let it boil for 2 minutes while stirring constantly.

Take the mixture off the heat, then incorporate butter and vanilla. Continuously stir until the mixture becomes smooth. Transfer the mixture into a medium-sized bowl. Use plastic wrap to cover the bowl, ensuring it touches the surface of the mixture directly. Place the bowl in the refrigerator and let it chill until it becomes cold.

To make the chocolate glaze, warm the cream either in a microwave or on medium heat until it is very hot but not boiling. Next, pour the hot cream over the chocolate chips and allow it to rest for 1 minute. Lastly, mix everything together vigorously until the chocolate is completely melted.

Load a pastry bag equipped with a circular nozzle with the cream. Employ a small knife to create a cavity in the side of each donut, penetrating approximately 75% into the donut's interior, then proceed to inject a substantial quantity of vanilla cream into the donut's core.

Dip the upper part of the donut into the glaze and let it rest for approximately 10 minutes before you serve it.