Step 1: Make the Donut Dough
In a small bowl, mix the yeast, warm milk, and 1 tablespoon of sugar. Let sit for 5–10 minutes until frothy.
In a large mixing bowl, combine the flour, remaining sugar, and salt. Add the yeast mixture, melted butter, and egg. Mix until a soft dough forms.
Knead the dough on a floured surface for about 5 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise for 1–2 hours, or until doubled in size.
Step 2: Prepare the Custard Filling
In a medium saucepan, heat the milk over medium heat until warm but not boiling.
In a bowl, whisk together the sugar, egg yolks, and cornstarch. Gradually pour the warm milk into the egg mixture, whisking constantly.
Return the mixture to the saucepan and cook over medium heat, stirring constantly, until thickened.
Remove from heat, stir in the vanilla extract, and let cool completely.
Step 3: Shape and Fry the Donuts
Roll out the dough to 1/2-inch thickness on a floured surface. Use a donut cutter or round cutter to cut out circles.
Heat vegetable oil in a deep pot to 350°F (175°C). Fry the donuts in batches for 1–2 minutes on each side until golden brown.
Transfer to a wire rack to drain and cool.
Step 4: Make the Chocolate Glaze
In a microwave-safe bowl, heat the heavy cream until hot. Pour over the chocolate chips and let sit for 2 minutes. Stir until smooth.
Step 5: Assemble the Donuts
Use a piping bag fitted with a round tip to fill each donut with the cooled custard.
Dip the top of each donut into the chocolate glaze. Place on a wire rack and let the glaze set.