Activate the Yeast: In a small bowl, combine the warm milk and active dry yeast. Allow the mixture to rest for 5 minutes, or until it becomes frothy.

Mix the Dough: In a large mixing bowl, combine the flour, sugar, and salt. Incorporate the yeast mixture, eggs, and softened butter. Stir until a soft dough comes together. Knead the dough by hand or using a stand mixer with a dough hook for 5–7 minutes, until it’s smooth and elastic.

First Rise: Transfer the dough to a lightly greased bowl, cover with a clean towel, and let it rise in a warm place for about 1 hour, or until doubled in size.

Heat the Milk: In a medium saucepan, heat the milk over medium heat until it just begins to simmer, then remove from heat.

Prepare the Custard Base: In a separate bowl, whisk together the sugar, egg yolks, and cornstarch until smooth. Gradually add the hot milk to the egg mixture while whisking continuously to prevent the eggs from curdling.

Thicken the Custard: Return the mixture to the saucepan and cook over medium heat, whisking constantly until the custard thickens and starts to bubble, about 2–3 minutes. Take the mixture off the heat and stir in the vanilla extract. Pour the custard into a bowl, cover with plastic wrap to avoid a skin forming, and refrigerate until fully chilled.

Shape the Donuts: Once the dough has risen, punch it down and roll it out on a lightly floured surface to about 1/2-inch thickness. Use a donut cutter or a round cutter (about 3 inches) to cut out donut rounds.

Second Rise: Place the donuts on a parchment-lined baking sheet, cover, and let them rise for about 30 minutes until puffy.

Fry the Donuts: Heat oil in a deep fryer or heavy-bottomed pot to 350°F (175°C). Cook the donuts in small batches for 1–2 minutes on each side, or until they turn golden brown. Use a slotted spoon to lift them out and place them on a plate lined with paper towels to drain excess oil.

Fill with Custard: Once the donuts have cooled, use a piping bag fitted with a round tip to fill each donut with the vanilla custard.

Caramelize the Sugar: Sprinkle a layer of granulated sugar on top of each filled donut. Use a kitchen torch to caramelize the sugar until golden and crispy, creating the iconic crème brûlée topping.