In a medium saucepan, whisk together milk, sugar, and cornstarch until smooth. Cook over medium heat, whisking constantly, until the mixture begins to steam and thicken slightly, about 4-5 minutes.

In a small bowl, whisk the egg yolks. Slowly pour in half the hot milk mixture while whisking constantly to temper the yolks. Return everything to the pot and cook, whisking, until thick and bubbling, about 2 minutes. Remove from heat, stir in butter and vanilla until smooth. Press plastic wrap directly onto the surface and refrigerate until cold, at least 1 hour.

In a large bowl, whisk together flour, sugar, baking powder, and salt. In a separate bowl, whisk milk, egg, and melted butter. Pour wet ingredients into dry and stir just until a sticky dough forms. Do not overmix.

Heat oil to 350°F in a heavy pot. Using two spoons, drop walnut-sized portions of dough into the oil, frying 4-5 at a time. Fry until deep golden brown and cooked through, 3-4 minutes total, turning occasionally. Transfer to a wire rack to cool completely.

Once cooled, use a small knife or skewer to poke a hole in each donut bite. Transfer cold custard to a piping bag fitted with a small round tip. Pipe custard into each bite until you feel slight resistance and the donut feels full.

Combine chocolate, heavy cream, and corn syrup in a microwave-safe bowl. Microwave in 30-second bursts, stirring between each, until smooth and glossy, about 1-1.5 minutes total.

Dip the top of each filled donut bite into the warm glaze, letting excess drip off. Place back on the rack, glaze side up, and let set for 15 minutes before serving.