In the lukewarm milk, dissolve the sugar, then crumble in the yeast and let it stand for 10 minutes, until foamy.
In a large bowl, sift the flour, add the salt, then pour in the yeast mixture and the vanilla extract. Start kneading the dough.
Once the dough comes together, place it on a lightly floured surface and knead for 10 minutes, until smooth and elastic.
Place the dough in a bowl, cover with a clean kitchen towel, and let it rise for 1 hour, or until doubled in size.
Roll out the risen dough into a rectangle. Place the cold butter in the center. Fold the edges of the dough over the butter to enclose it completely.
Gently roll out the dough, then fold it into thirds like a letter. Cover and refrigerate for 30 minutes.
Repeat the folding and rolling process two more times, always chilling the dough for 30 minutes in the refrigerator between each fold.
Roll out the prepared puff pastry to a thickness of about 5 mm. Cut into triangles and sprinkle with vanilla sugar.
Roll up the triangles starting from the wide end towards the tip to form croissant shapes.
Place the croissants on a baking sheet lined with parchment paper, cover, and let rise for another 30 minutes.
Whisk the egg and brush the tops of the croissants with it to give them a nice shine.
Bake in a preheated oven at 375°F (190°C) for 15-18 minutes, until golden brown.
Let the croissants cool down, then sprinkle with powdered sugar before serving.
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