In lukewarm milk, stir in the sugar, then crumble in the yeast. Let it sit for 10 minutes until frothy.
In a large bowl, sift the rye flour and all-purpose flour, add the salt and ground caraway seeds, then rub in the cold butter until the mixture resembles breadcrumbs.
Pour in the proofed yeast mixture, sour cream, and whole egg. Knead the dough until it becomes elastic and smooth.
Mix in the grated cheese, then cover the dough and let it rise in a warm place for about 1 hour.
On a lightly floured surface, roll out the dough to about ¾ inch (2 cm) thickness, then fold it over a few times to create layers.
Roll out the dough again to ¾ inch (2 cm) thickness, then use a round cookie cutter to cut out the scones. Place them on a baking sheet lined with parchment paper.
Brush the tops with beaten egg, then bake in a preheated oven at 350°F (180°C) until golden brown, about 20-25 minutes.
Let the scones cool on a wire rack, then serve fresh or store in an airtight container.
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