In lukewarm milk, dissolve the sugar, then crumble in the fresh yeast. Let it stand for 10 minutes, until foamy.
In a large bowl, sift the flour and add the salt, then rub in the cold butter until the mixture resembles breadcrumbs.
Pour in the foamy yeast mixture, sour cream, and the whole egg. Knead the dough until it becomes elastic.
Mix half of the grated cheese into the dough, then cover and let it rise in a warm place for about 1 hour.
On a floured surface, roll out the dough to about ¾ inch (2 cm) thickness, then fold it several times to create flaky layers in the pogácsa.
Roll out the dough again to ¾ inch (2 cm) thickness, then use a pogácsa cutter or a round 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 the remaining grated cheese, then bake in a preheated oven at 400°F (200°C) until golden brown, about 20-25 minutes.
Let the pogácsa cool on a wire rack to make them even crispier, then serve fresh or store in an airtight container.
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