Make the Pastry Cream : In a medium saucepan, heat the milk over medium heat until steaming but not boiling. In a separate bowl, whisk together the egg yolks, sugar, and cornstarch until smooth. Slowly pour the hot milk into the egg mixture while whisking constantly. Return the mixture to the saucepan. Cook over medium heat, whisking, until thickened (about 2-3 minutes). Remove from heat and stir in the butter and vanilla extract. Transfer to a bowl, cover with plastic wrap (pressing it directly onto the surface to prevent a skin from forming), and refrigerate until fully chilled.

Prepare the Yeast Donut Dough : In a large bowl, mix the lukewarm milk, 1 tablespoon of the sugar, and yeast. Let sit for 5-10 minutes until frothy. Add the remaining sugar, egg, vanilla, melted butter, flour, and salt. Mix until a dough forms. Knead the dough on a floured surface for 8-10 minutes or until smooth and elastic. Alternatively, use a stand mixer with a dough hook for 5-7 minutes. Place the dough in a greased bowl, cover, and let rise in a warm spot for 1-1½ hours, or until doubled in size.

Shape and Bake the Donuts : Punch down the dough and roll it out to about ½ inch thickness. Use a 3-inch round cutter to cut out donuts. Place the rounds on a parchment-lined baking sheet, cover lightly, and let rise for another 30 minutes. Preheat the oven to 375°F (190°C). Brush the tops of the donuts with vegetable oil and bake for 10-12 minutes, or until lightly golden.

Fill the Donuts : Allow the donuts to cool completely. Use a piping bag with a small round tip to fill each donut with the chilled pastry cream.

Caramelize the Sugar Topping : In a saucepan, combine the sugar, water, and salt. Cook over medium heat without stirring until the sugar melts and turns a deep amber color. Carefully drizzle the caramelized sugar over the tops of the filled donuts, or dip the tops directly into the sugar for an even coating. Let the sugar harden.