Make the Pastry Cream (Do This First):

1. In a medium saucepan, heat the milk over medium heat until it just begins to steam and small bubbles form around the edges. Do not boil.
2. Meanwhile, whisk together egg yolks, sugar, and cornstarch in a medium bowl until smooth and pale yellow, about 2 minutes.
3. Slowly pour about ½ cup of the hot milk into the egg mixture, whisking constantly to temper the eggs. Then pour the egg mixture back into the saucepan with the remaining milk, whisking constantly.
4. Cook over medium heat, whisking constantly, until the mixture thickens and comes to a boil, about 3-5 minutes. Let it boil for 1 minute while continuing to whisk.
5. Remove from heat and whisk in butter and vanilla extract until smooth. Strain through a fine-mesh sieve into a clean bowl to remove any lumps.
6. Press plastic wrap directly onto the surface of the pastry cream to prevent a skin from forming. Refrigerate for at least 2 hours until completely chilled.
7. Before using, whip the cold heavy cream to stiff peaks and gently fold it into the chilled pastry cream for an extra-light filling. Transfer to a piping bag fitted with a round tip.

Make the Donut Dough:

8. In the bowl of a stand mixer, combine 1 cup flour, sugar, and yeast. Add the warm milk and stir to combine. Let stand for 5 minutes until foamy.
9. Add eggs, butter, salt, and vanilla extract. Mix on low speed until combined.
10. Gradually add remaining 2 ¾ cups flour, mixing on low speed until a soft dough forms. Increase to medium speed and knead for 5-7 minutes until the dough is smooth and elastic (it should be slightly sticky but pull away from the sides of the bowl).
11. Place dough in a greased bowl, cover with a clean kitchen towel, and let rise in a warm place for 1-1.5 hours, or until doubled in size.

Shape and Fry the Donuts:

12. Gently punch down the dough and turn it out onto a lightly floured surface. Roll to about ½-inch thickness.
13. Using a 3-inch round cutter, cut out circles. Place them on parchment-lined baking sheets, spacing them 2 inches apart. Cover with towels and let rise for 30-45 minutes until puffy.
14. Heat oil in a large, heavy-bottomed pot to 350°F (use a thermometer to monitor temperature).
15. Carefully add 2-3 donuts at a time to the hot oil. Fry for 1-2 minutes per side until golden brown. Don’t overcrowd the pot.
16. Remove with a slotted spoon and drain on a wire rack set over paper towels. Let cool completely before filling.

Make the Chocolate Ganache:

17. Place chocolate chips in a heat-proof bowl. Heat heavy cream in a small saucepan until it just begins to simmer.
18. Pour hot cream over chocolate chips and let sit for 1 minute. Add butter and corn syrup (if using), then whisk until smooth and glossy.

Assemble the Donuts:

19. Using a skewer or chopstick, poke a hole in the side of each cooled donut and wiggle it around to create a pocket for the filling.
20. Insert the piping bag tip into each hole and pipe pastry cream until you feel slight resistance (the donut will feel heavier and slightly fuller).
21. Dip the top of each filled donut into the chocolate ganache, allowing excess to drip off. Place on a wire rack and let the chocolate set for 10-15 minutes.
22. Serve immediately or store properly. These are best enjoyed within 24 hours!