Activate the yeast: In lukewarm milk, mix in the sugar and yeast. Let it sit for 10 minutes until foamy.
In a large bowl, combine the flour and salt. Cut in the butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs.
Grate the smoked cheese and add half of it to the flour mixture.
Add the sour cream, egg, and activated yeast to the bowl. Knead until a smooth dough forms.
On a lightly floured surface, roll out the dough to about ¾ inch (2 cm) thickness. Fold the dough in thirds, like a letter, to create layers.
Cover with a clean kitchen towel and let rest for 30 minutes to rise.
Roll out the dough again to about ¾ inch (2 cm) thickness and use a round cookie cutter to cut out the pogácsa shapes.
Place the pogácsa on a baking sheet lined with parchment paper and brush the tops with beaten egg.
Sprinkle the remaining smoked cheese on top, then bake in a preheated oven at 350°F (180°C) for 20-25 minutes, or until golden brown.
Let them cool slightly before serving. Enjoy fresh, with a soft and crumbly texture.
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