1. Make the Pastry Cream Filling

1. Heat the milk in a medium saucepan over medium heat until steaming with bubbles forming at the edges. Remove from heat.
2. In a medium bowl, whisk together egg yolks, sugar, and cornstarch until smooth.
3. Slowly pour the warm milk into the egg mixture, whisking constantly to temper the eggs.
4. Return the mixture to the saucepan and cook over medium heat, whisking constantly, until thickened to a pudding-like consistency (5–10 minutes).
5. Remove from heat and whisk in butter and vanilla. Transfer to a heat-safe bowl, cover with plastic wrap touching the surface, and refrigerate for 1–2 hours until set.

2. Prepare the Donut Dough

1. In a large bowl, whisk together lukewarm milk, 1 tablespoon sugar, and yeast. Let sit for 5–10 minutes until foamy.
2. Add the remaining sugar, egg, vanilla, and melted butter to the yeast mixture and whisk to combine.
3. Gradually mix in half the flour, stirring to form a rough dough, then add the remaining flour and salt. Mix until a sticky dough forms.
4. Knead the dough on a floured surface for 5–10 minutes until smooth and elastic.
5. Grease a large bowl with vegetable oil, place the dough inside, and turn to coat. Cover and let rise in a warm place for 45 minutes to 1 hour until doubled in size.

3. Shape and Bake the Donuts

1. Punch down the dough and roll it out to ½-inch thickness on a lightly floured surface. Use a 3-inch round cutter to shape the donuts.
2. Arrange the cut donuts on parchment-lined baking sheets, leaving space between each. Let rise for 30–45 minutes until doubled in size.
3. Preheat your oven to 375°F. Bake the donuts for 10–12 minutes, until golden brown and puffed. Cool completely on a wire rack.

4. Fill the Donuts

1. Use a sharp knife to create a small hole in the side of each donut.
2. Fill a piping bag with the chilled pastry cream and pipe into each donut until cream peeks out.

5. Caramelize the Sugar

1. In a medium skillet, combine sugar, water, and salt. Cook over medium-high heat, swirling the pan gently (do not stir), until the sugar turns amber in color.
2. Remove from heat and let sit for 1 minute. Dip the tops of the filled donuts into the caramel, letting excess drip off.
3. Place the donuts on a wire rack or parchment paper to cool and harden the sugar topping.