In a small bowl, combine warm milk (about 110°F/43°C) and granulated sugar. Sprinkle yeast over the top and let it bloom for 5-10 minutes until foamy.

In a large bowl of a stand mixer fitted with the dough hook, combine flour and salt. Add the bloomed yeast mixture, eggs, and softened butter. Mix on low speed until a shaggy dough forms.

Increase speed to medium-low and knead for 8-10 minutes, or until the dough is smooth and elastic.

Grease a clean bowl. Place the dough in the bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1-1.5 hours, or until doubled in size.

In a medium saucepan, heat milk over medium heat until steaming but not boiling.

In a separate bowl, whisk together egg yolks, sugar, and cornstarch until smooth.

Slowly temper the egg yolk mixture by whisking in about half of the hot milk. Pour the tempered egg mixture back into the saucepan with the remaining milk.

Cook over medium-low heat, whisking constantly, until the custard thickens and bubbles. Remove from heat and stir in vanilla extract.

Pour custard into a bowl, cover the surface directly with plastic wrap to prevent a skin from forming, and refrigerate until completely chilled.

Punch down the risen dough and turn it out onto a lightly floured surface. Roll out the dough to about 0.5-inch thickness.

Cut out donuts using a donut cutter or two round cutters of different sizes.

Place donuts on parchment-lined baking sheets, cover loosely with plastic wrap, and let them rise for another 30-45 minutes until puffy.

Heat vegetable oil in a Dutch oven or deep skillet to 350°F (175°C).

Carefully fry donuts in batches for 2-3 minutes per side, until golden brown. Drain on paper towels.

In a heatproof bowl set over a saucepan of simmering water (double boiler), melt chocolate chips, butter, and corn syrup, stirring until smooth.

Once donuts are cool enough to handle, use a piping bag or a sharp knife to make a hole in the side of each donut. Fill with chilled custard.

Dip the top of each filled donut into the chocolate glaze. Let the glaze set before serving.