Activate the yeast: in lukewarm milk, mix in the sugar and yeast, then let it stand for 10 minutes until foamy.
In a large bowl, combine the flour and salt, then rub in the butter until evenly distributed and resembles breadcrumbs.
Finely chop half of the fresh rosemary, and mix it into the flour mixture along with half of the grated cheese.
Add the sour cream, egg, and activated yeast, then knead the dough thoroughly until it comes together.
On a floured surface, roll out the dough to about 2 cm (¾ inch) thick, then fold it three times to create a layered texture.
Cover with a kitchen towel and let it rest for 30 minutes to rise slightly.
Roll out the dough again to about 2 cm (¾ inch) thick, then use a pogácsa cutter (or round cookie cutter) to cut out the shapes.
Place the pogácsa on a baking sheet lined with parchment paper, and brush the tops with beaten egg.
Sprinkle the remaining grated cheese and rosemary on top, then bake in a preheated oven at 180°C (350°F) for 20-25 minutes, or until golden brown.
Let them cool slightly, then serve fresh, soft, and crumbly.
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