Make the dough: Combine warm milk, yeast, and 1 tsp sugar in a bowl. Let sit until foamy. Add remaining sugar, eggs, butter, flour, salt, and vanilla. Knead until smooth and elastic. Cover and let rise until doubled, 1 to 1.5 hours.

1. Shape donuts: Roll dough to 1/2 inch thick. Cut into rounds with a cutter. Place on a floured surface, cover, and let rise 30–45 minutes.
2. Fry: Heat oil to 350°F (175°C). Fry donuts until golden brown on both sides. Drain on paper towels. Cool completely.
3. Make pastry cream: Heat milk in a saucepan until steaming. In a bowl, whisk yolks, sugar, and cornstarch. Gradually whisk in hot milk. Return to saucepan, cook over medium heat until thickened. Remove from heat, stir in butter and vanilla. Chill until firm.
4. Fill donuts: Place pastry cream in a piping bag with a small tip. Insert into each donut and fill generously.
5. Brûlée topping: Sprinkle granulated sugar over the tops. Use a kitchen torch to caramelize until golden and crisp. Let cool briefly before serving.