Place the flour in a large bowl. Dissolve the yeast in warm water and add it to the flour. Add sugar, salt, and start kneading the dough, gradually adding the warm milk. Cover the dough with a cloth and set it aside in a warm place for 1.5 hours.
In the meantime, crumble the feta cheese and grate the halloumi and mozzarella.
Transfer the risen dough to a countertop and divide it into four parts. Roll out a portion of the dough and sprinkle with mozzarella. Fold the edges of the dough, forming a boat shape.
Place the khachapuri on a baking sheet lined with parchment paper. Fill the centre with halloumi and feta, and brush the dough with a beaten egg. Put it in the oven preheated to 220°C for 10 minutes.
After this time, remove the khachapuri from the oven and crack an egg in the centre, being careful not to break the yolk. Add a piece of butter and put it back in the oven for another 5 minutes. After that, serve the khachapuri, sprinkling it with chives.