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In a bowl, mix the warm milk, yeast, and a pinch of sugar. Let sit for 5–10 minutes until frothy.
In a large mixing bowl, combine the flour, sugar, and salt. Add in the yeast mixture, egg, softened butter, and vanilla extract.
Knead until a smooth, elastic dough forms (8–10 minutes by hand or 5–6 minutes in a stand mixer).
Cover and let rise in a warm place for 1 hour, or until doubled in size.
In a saucepan, whisk together milk, sugar, and cornstarch. Add egg yolks and whisk until smooth.
Cook over medium heat, stirring constantly, until the mixture thickens (about 5–7 minutes).
Remove from heat and stir in butter and vanilla. Let cool completely, then refrigerate.
Roll out the dough to about ½ inch (1.2 cm) thick. Use a round cutter (about 2.5–3 inches) to cut circles.
Place on a floured surface, cover, and let rise for another 30 minutes.
Heat oil in a deep pot to 340–350°F (170–180°C). Fry donuts in batches, 1–2 minutes per side, until golden brown.
Drain on paper towels and let cool slightly.
Once cool, use a piping bag fitted with a long nozzle to inject the vanilla cream into the side of each donut.
Dust generously with powdered sugar.
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