01 - In the bowl of a stand mixer, combine the all-purpose flour, granulated sugar, active dry yeast, and fine sea salt. Add lukewarm whole milk and large eggs. Mix on low speed until ingredients begin to combine. Gradually incorporate softened unsalted butter, one tablespoon at a time. Continue mixing until the butter is fully blended into the dough.
02 - Knead the dough on medium speed for 10 minutes, until it becomes smooth and elastic. Transfer the dough to a greased bowl, cover, and let rise in a warm place until doubled in size, about 1 to 1 and 1/2 hours.
03 - On a lightly floured surface, roll the dough to 1/2-inch thickness. Cut out 3-inch rounds using a cookie cutter. Arrange on parchment-lined trays, cover, and allow to rise for 30 minutes until slightly puffy.
04 - Heat vegetable oil in a deep fryer or heavy pot to 350°F. Fry donuts in batches, 2 to 3 minutes per side, until golden brown. Remove with a slotted spoon and place on paper towels to drain. Let donuts cool completely.
05 - Heat whole milk in a saucepan until steaming but not boiling. In a separate bowl, whisk together egg yolks, granulated sugar, cornstarch, and a pinch of salt until smooth. Gradually pour the hot milk into the egg yolk mixture while whisking constantly. Return mixture to saucepan and cook over medium heat, whisking continuously, until thickened, about 2 to 3 minutes. Remove from heat; stir in unsalted butter and vanilla. Transfer to a bowl, press plastic wrap directly onto the surface, and chill until cold.
06 - Fill a piping bag fitted with a medium round tip with the chilled custard. Poke a hole in the side of each donut and pipe in vanilla custard until the donut feels slightly full.
07 - Dip or sprinkle each donut top with a thin, even layer of granulated sugar. Using a kitchen torch, caramelize the sugar until it is deeply golden and forms a crisp, crackling crust. Allow to cool briefly before serving.