Add 1 teaspoon of sugar over the yeast and ⅓ cup of the milk. Stir to dissolve.
In a large bowl combine flour, rest of the sugar, salt. Add melted butter, dissolved yeast, eggs and the rest of the milk.
Stir to combine and knead until the dough is soft, a bit sticky but quite elastic and pulls away from the side of the bowl. If it feels too soft, you can add extra flour and if it is not soft enough add a bit milk
Cover the dough with a plastic wrap, place it in a warm place and let it rise for about 40min or until double in size.
To make the cheese filling
In a medium bowl combine the cream cheese with the feta cheese, mozzarella and one egg.
To Assemble
Remove the dough from the bowl and place it onto a floured parchment paper(easier to do it over parchment paper, because the gluten-free dough is very fragile and this way you won't need to transfer it and risk breaking it in the process). Then cut it into 2 equal pieces. (for smaller bread cut into 4 equal pieces)
Spread each piece of the dough into a circle. Then roll 2 opposite sides of the circle towards the center so it ends up have a boat like shape. Then pinch the corners together.
Stuff each khachapuri with the cheese mixture. Beat 1 egg with a teaspoon of water, then brush the dough with egg wash.
Bake in a preheated 450°F (230°C ) oven for about 15 minutes or until the crust becomes golden brown.
Make a well in the center of each khachapuri with the back of a spoon (about 3 inches in diameter) and drop 1 egg into each well. Then stick a few small pieces of butter into the cheese.
Return the khachapuri back into the oven and bake for another 5-6 minutes. Cooking time may vary depending on your oven. The egg white should be white but still pretty runny. It will cook further as it sits in the hot cheese.