Heat the milk for the pastry cream in a medium saucepan over medium heat until steaming and just beginning to bubble around the edges. Remove from the heat.

In a bowl, whisk the egg yolks, sugar, and cornstarch until smooth. Slowly pour in the hot milk while whisking constantly.

Strain the mixture back into the saucepan. Cook over medium heat, whisking constantly, until thickened to a soft pudding consistency.

Remove from the heat and whisk in the butter and vanilla extract.

Transfer the pastry cream to a heat-safe bowl and press plastic wrap directly onto the surface. Chill for 1 to 2 hours until fully cold and set.

For the donut dough, warm the milk until just warm to the touch, not hot.

In a large bowl, whisk together the warm milk, 1 tablespoon of the sugar, and the yeast. Let it sit for 5 to 10 minutes until foamy.

Add the remaining sugar, egg, vanilla, and melted butter. Whisk to combine.

Mix in half the flour, then add the remaining flour and salt. Stir until a rough dough forms.

Turn the dough onto a floured surface and knead for 5 to 10 minutes until smooth and elastic.

Place the dough in an oiled bowl, cover, and let rise in a warm place for 45 minutes to 1 hour, or until doubled.

Line 2 baking sheets with parchment paper.

Punch down the dough and roll it out to about 1/2 inch thick. Cut out rounds with a 3-inch cutter.

Place the rounds on the prepared baking sheets, leaving space between them. Reroll scraps as needed.

Cover and let the donuts rise again for 30 to 45 minutes until puffy and doubled.

Preheat the oven to 375°F.

Bake the Baked Crème Brûlée Donuts for 10 to 12 minutes until lightly golden.

Transfer to a wire rack and cool completely.

Use a small sharp knife to make a hole in the side of each donut. Pipe the chilled pastry cream inside each one.

Refrigerate the filled donuts while you prepare the sugar topping.

In a skillet, combine the sugar, water, and salt. Bring to a boil over medium-high heat without stirring, swirling gently if needed.

Cook until the sugar turns a light honey color, then remove from the heat. Let it deepen slightly to amber from residual heat.

Dip the tops of the filled donuts into the caramelized sugar using tongs.

Place on a wire rack or parchment paper and let the topping harden for 5 to 10 minutes before serving.