Making the Pastry Cream Filling: Pour milk into a medium saucepan and heat until steaming. In a bowl, whisk egg yolks, sugar, and cornstarch until smooth. Gradually whisk the hot milk into the egg mixture to temper the eggs. Return the mixture to the saucepan and cook over medium heat, stirring constantly, until thickened. Off the heat, whisk in butter and vanilla. Transfer to a bowl, cover with plastic wrap, and chill for 1–2 hours.

Preparing the Donut Dough: Warm the milk until lukewarm (110°F). Mix with 1 tablespoon of sugar and yeast. Let sit for 5–10 minutes until foamy. Add the remaining sugar, egg, vanilla, and melted butter. Mix until combined. Gradually add the flour and salt, mixing until a dough forms. Knead on a floured surface for 5–10 minutes until smooth and elastic. Place dough in an oiled bowl, cover, and let rise until doubled (about 1 hour). Roll the dough to 1/2-inch thickness and cut with a 3-inch round cutter. Arrange donuts on parchment-lined baking sheets, cover, and let rise for another 30–45 minutes. Preheat oven to 375°F and bake donuts for 10–12 minutes, until golden.

Caramelizing the Sugar Topping: Heat sugar, water, and salt in a skillet over medium-high heat until amber-colored. Quickly dip the tops of the donuts into the caramel using tongs. Let cool on a wire rack until the caramel hardens.