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