Mix the flour and salt in a bowl. Add the yeast, milky water and oil, mix with a wooden spoon. Knead to form a soft, smooth dough. Cover and leave to rise at room temperature for approx. 1 hr. until doubled in size.
Mix the feta with the drained liquid and mozzarella.
On a lightly floured surface, cut the dough into quarters and roll out into ovals. Spread the cheese mixture in the centre of the pieces of dough. Fold the longer edges approx. 3 cm in towards the middle, ensuring the cheese is still visible. Twist the ends, place the khachapuri on two baking trays lined with baking paper.
Approx. 17 mins. per tray on the bottom shelf of an oven preheated to 240°C. Remove from the oven, crack the eggs one at a time and slide into the centre of each khachapuri. Carefully mix the egg white with the cheese, making sure the yolk remains whole. Return the khachapuri to the upper half of the oven for approx. 3 mins., then remove. Brush the edges with butter while still hot, serve immediately. Repeat the process with the second tray.