Whisk the egg yolks, sugar, cornstarch, and salt in a bowl. Heat the oat milk until steaming, gradually whisk it into the yolks, and return the mixture to the saucepan.
Cook the custard over medium heat, whisking continuously, until thick and bubbling. Cook for 1 more minute, stir in the plant butter and vanilla, then cover the surface and refrigerate until cold.
Combine the warm oat milk, yeast, and 1 tablespoon of the doughnut sugar. Let it stand for 5 to 10 minutes until foamy, then mix in the remaining sugar, eggs, and melted plant butter.
Add the flour and salt and knead for 8 minutes, until the dough is smooth and elastic. Cover and let it rise in a warm place for 60 to 75 minutes, until doubled.
Roll the dough to 1/2-inch thickness and stamp out 12 rounds with a 3-inch round cutter. Place the rounds on parchment paper, cover, and let them rise for 30 to 40 minutes until puffy.
Heat the neutral oil in a deep, heavy pot to 350 F. Fry the doughnuts in small batches for about 90 seconds per side, adjusting the heat to maintain the temperature.
Drain the doughnuts briefly on a rack, then coat them in granulated sugar while still warm. Let them cool completely before filling.
Make a small opening in the side of each doughnut with a piping tip. Pipe the chilled vanilla custard into the centers and serve the doughnuts the same day.