Whisk warm milk, yeast, and 1 tablespoon sugar in a bowl. Let sit 5 minutes until foamy - if nothing happens, your yeast is dead and you need to start over.

In a stand mixer bowl, combine bread flour, remaining 2 tablespoons sugar, and salt. Add the yeast mixture, softened butter, and egg. Mix on low until a shaggy dough forms, then knead on medium 8-10 minutes until smooth and elastic. The dough should pass the windowpane test - stretch a small piece thin enough to see light through without tearing.

Transfer to a lightly oiled bowl, cover with plastic wrap, and let rise in a warm spot 60-75 minutes until doubled in size. Press a finger in - the indentation should stay.

Punch down dough and divide into 8 equal pieces (about 3 ounces each). Roll into tight balls, place on parchment-lined baking sheets with plenty of space between them, cover loosely, and rise 30-40 minutes until puffy and nearly doubled. They should look like soft marshmallows.

While dough rises, whisk milk, cornstarch, sugar, and egg yolks in a small saucepan until smooth. Cook over medium heat, whisking constantly, until mixture thickens and bubbles - about 4-5 minutes. It should coat the back of a spoon and hold a line when you drag your finger through.

Remove from heat, whisk in cold butter and vanilla until glossy. Press plastic wrap directly onto surface and refrigerate until cold, at least 30 minutes. Whip heavy cream to stiff peaks, then fold into chilled pastry cream. Transfer to piping bag and refrigerate until ready to fill.

Heat oil to 350 degrees F in a deep pot. Carefully lower in 2-3 donuts at a time, parchment and all - the paper will release and you can fish it out. Fry 90 seconds per side until deep golden brown. Drain on a wire rack set over a baking sheet.

While warm, roll donuts in sugar to coat completely. Let cool 10 minutes. Poke a deep hole in the side of each donut with a chopstick or skewer, wiggling to create space. Pipe in milk cream until the donut feels heavy and the cream nearly overflows - you'll use 3-4 tablespoons per donut. Serve immediately or within 4 hours.