Step 1: Prepare the Pastry Cream

1. In a saucepan, heat the milk over medium heat until steaming but not boiling.
2. In a bowl, whisk together egg yolks, sugar, and cornstarch until smooth.
3. Slowly pour the hot milk into the yolk mixture, whisking constantly. Return the mixture to the saucepan and cook until thickened.
4. Stir in butter and vanilla, then refrigerate the cream with plastic wrap pressed directly on the surface to avoid a skin.

Step 2: Make the Dough

1. Mix lukewarm milk, 1 tablespoon of sugar, and yeast. Let sit until frothy.
2. Whisk the egg, remaining sugar, vanilla, and melted butter, and combine with the yeast mixture.
3. Gradually add flour and salt, mixing to form a dough. Knead on a floured surface for 5–7 minutes.
4. Place the dough in a greased bowl, cover, and let rise for 1–2 hours until doubled in size.

Step 3: Shape and Bake Donuts

1. Roll the dough to ½-inch thickness and cut out circles using a donut cutter or glass.
2. Arrange the donuts on a parchment-lined tray and let rise for 30 minutes.
3. Preheat oven to 350°F (175°C) and bake for 10–12 minutes. Let cool completely.

Step 4: Caramelize the Sugar

1. Combine sugar, water, and salt in a saucepan over medium heat. Stir until dissolved, then boil without stirring until golden.
2. Dip the tops of the donuts into the caramel and let cool.

Step 5: Fill the Donuts

1. Pipe the chilled pastry cream into the center of each donut using a piping bag.
2. Serve immediately and enjoy the crispy caramelized top with the creamy filling.