Clean the dock and cut it up. Saute it with the oil until it softens. Let it cool and add the crumbled feta cheese. Stir.
Divide the risen dough into 2 balls. Form boat shapes from each ball of dough. Fill them with the stuffing of dock and feta cheese. Sprinkle with grated cheese.
Make a well in the middle of each Khachapuri and break an egg into each. Put the Khachapuri in a buttered oven dish and bake until golden.