Preparing the Dough:

Activate the Yeast:

Combine the lukewarm milk, yeast, and a teaspoon of sugar in a large mixing bowl. Allow the mixture to sit for about 10-15 minutes until it becomes foamy, indicating that the yeast is active.

Mix the Dough Ingredients:

Add the egg yolks and vanilla extract to the yeast mixture, stirring well to combine.

In the bowl of a stand mixer fitted with a dough hook, combine the flour, remaining sugar, butter, and salt.

Pour in the yeast mixture and mix on low speed until the ingredients are just combined, then increase to medium speed and knead until the dough is smooth and elastic, about 5-10 minutes.

First Rise:

Transfer the dough to a lightly greased bowl, cover with plastic wrap or a clean kitchen towel, and let it rise in a warm, draft-free place for 1-2 hours or until doubled in size.

Making the Custard:

Prepare the Custard Base:

In a medium bowl, whisk together the sugar, salt, and cornstarch.

Beat the egg yolks until smooth, then gradually mix in the dry ingredients until the mixture is pale yellow and smooth.

Heat the milk in a saucepan over medium-high heat just until it starts to simmer. Gradually temper the hot milk into the egg mixture, stirring continuously to prevent the eggs from scrambling.

Cook the Custard:

Return the mixture to the saucepan and cook over medium heat, stirring constantly, until the custard thickens and begins to bubble.

Remove from heat and stir in the vanilla extract and butter until the custard is smooth.

Transfer the custard to a bowl, cover with plastic wrap pressed directly onto the surface to prevent a skin from forming, and refrigerate until cool.

Frying and Assembling the Donuts:

Shape and Fry the Donuts:

Punch down the risen dough and turn it out onto a lightly floured surface. Roll the dough to about 1/2-inch thickness.

Use a 3-inch cookie cutter to cut out rounds. Place these on a prepared baking sheet, cover, and let them rise again until puffed, about 20-30 minutes.

Heat the vegetable oil in a deep fryer or large pot to 360°F. Fry the donuts in batches until golden brown, about 1-2 minutes per side. Remove and drain on paper towels.

Fill and Finish with Brûlée:

Fill a piping bag with the custard and inject each donut.

For the brûlée topping, cook the sugar and water in a saucepan until golden. Dip the top of each filled donut into the caramel, then allow to set until hard.