Make the pastry cream first so it has time to chill: in a medium heavy-bottomed saucepan whisk together ¼ cup granulated sugar, 1 ½ tablespoons cornstarch, and a pinch of salt. Whisk in 1 1/4 cup whole milk and 1 large egg yolk until smooth.
Cook the pastry cream over medium heat, whisking constantly, until it thickens and comes to a bubble (about 5–8 minutes). Continue cooking and whisking for an additional 1–2 minutes. Remove from heat and whisk in 1 tablespoon unsalted butter and 1 teaspoon vanilla extract.
Transfer the pastry cream to a heat-proof container, press plastic wrap directly onto the surface to prevent a skin, and chill in the refrigerator for at least 4 hours or overnight.
Prepare the enriched dough: combine ½ cup heavy cream, ½ cup whole milk, ¼ cup granulated sugar (dough sugar), and ¼ cup unsalted butter in a heat-proof bowl or measuring cup. Heat in the microwave in 20-second intervals, stirring between intervals, until the butter is melted and the mixture reaches 130°F on an instant-read thermometer.
In the bowl of a stand mixer fitted with the dough hook, combine 3 ¼ cups all-purpose flour, 2 packets rapid rise yeast, and 1 teaspoon salt. Add the warm milk/butter mixture and mix on low just until the dough comes together.
Add 2 egg yolks to the mixer and mix on low until incorporated. Then knead on low speed with the dough hook for 5 minutes.
Turn the dough onto a lightly floured surface and hand-knead for 5–8 minutes, until the dough is smooth and springs back when gently poked.
Form the dough into a ball, place it in a large greased bowl, cover, and let rest for 20 minutes.
After the 20-minute rest, punch the dough down and roll it out on a lightly floured surface to ½-inch thickness. Use a 3-inch round cutter to cut out donuts. Re-roll scraps as needed.
Place the cut donuts on a parchment-lined baking sheet, cover loosely with a clean kitchen towel, and let rise at room temperature until nearly doubled in size, about 1 hour.
While the donuts are proofing, heat vegetable oil in a large, heavy-bottomed pot and use a candy thermometer to bring it to 360°F. Maintain the oil temperature between 350°F and 375°F while frying.
Fry the donuts in batches, careful not to overcrowd the pot. Fry about 2–3 minutes per side, until golden brown, turning once. Transfer fried donuts with a spider or slotted spoon to a cooling rack set over paper towels to drain.
Let the donuts cool until they are no longer hot to the touch (warm or room temperature) before filling.
Fill the donuts: transfer the chilled pastry cream to a piping bag fitted with a small round tip (or a zip-top bag with a small corner snipped). Insert the tip into the side or bottom of each donut and pipe the cream into the center until filled.
Make the chocolate ganache: heat 1/3 cup heavy whipping cream in the microwave for about 45 seconds, then pour it over 4 ounces semi-sweet chocolate chips in a heat-proof bowl. Let sit 2–3 minutes, then stir until smooth.
Dip the tops of the filled donuts into the ganache, allowing excess to drip off, and place back on the rack to let the chocolate set.
Serve once the chocolate is set. Store any leftovers refrigerated (pastry cream-filled donuts are best eaten within 1–2 days).