Combine the lukewarm milk, yeast, and 1 tsp of the measured sugar with a whisk, wait 10–15 minutes, then mix in the yolks and vanilla. A creamy foam layer should develop before the eggs go in, showing that the yeast is active.

Use a stand mixer fitted with its dough hook to combine the yeast mixture, flour, softened butter, salt, and remaining sugar, beginning on low and increasing to medium for 5–10 minutes of kneading. The finished dough should stretch readily and feel soft and slightly tacky rather than crumbly.

Move the dough into a large, lightly greased bowl, cover with plastic wrap or a clean towel, and leave in a warm spot away from drafts for 1–2 hours. Look for roughly double the starting volume before moving on.

Blend the sugar, salt, and cornstarch separately, then whisk them into smooth egg yolks in a medium bowl for about 1 minute. The mixture should become evenly combined and pale yellow.

Bring the milk just to a simmer over medium-high heat, take it off the burner, and gradually whisk ladlefuls into the yolks before returning everything to the saucepan over medium heat. Adding the milk slowly keeps the mixture smooth instead of creating cooked egg flecks.

Keep the whisk moving until the custard thickens and bubbles, then take the pan off the burner, incorporate the vanilla and butter, and scrape into a bowl. Slow, heavy bubbles breaking through the surface signal a thoroughly thickened mixture.

Lay plastic wrap directly against the custard and refrigerate for 1–2 hours, or until completely cold. Contact with the wrap prevents a skin, while chilling gives the filling its pipeable body.

Deflate the risen dough and roll it on a lightly floured work surface to about 1/2-inch thickness. A consistent thickness helps the rounds cook evenly.

Stamp out rounds with a 3-inch biscuit or cookie cutter, then gather, knead, and reroll the scraps to cut additional rounds. Keep the rounds intact rather than cutting center holes so they can hold the filling.

Arrange the rounds about 1 inch apart on a lightly greased baking sheet, cover with plastic wrap or a clean towel, and proof for 20–30 minutes. They should look puffy and approximately doubled before frying.

Bring vegetable oil to 360°F in a large pot, fry the rounds in batches for 1–2 minutes on the first side and about 1 minute after turning, then lift them with a slotted spoon onto paper towels. Both surfaces should be golden brown, with enough space around each donut to turn it comfortably.

Load the chilled custard into a piping bag fitted with a tip, make a cavity in each donut using a skewer or chopstick, and pipe through that opening. The donut should grow heavier and expand gently in your hand without splitting.

Heat the brûlée sugar and water in a small saucepan over medium-high heat, reduce to medium when bubbling begins, and gently swirl until golden before removing from the burner and swirling again. The color will deepen toward amber from residual heat, so stop before it becomes dark brown.

Protect your hands with suitable heat-resistant gloves, dip each donut’s top into the caramel, and lift and tilt it away from your fingers to drain excess before setting it down. A thin, glossy coating will firm as it cools.

Continue coating the remaining donuts, returning the pan briefly to gentle heat and swirling whenever the caramel becomes too stiff. It should flow freely enough to leave a thin cap rather than a heavy mound.