In a small bowl or liquid measuring cup, heat the milk to lukewarm. Add the yeast into the warm milk along with 1 tsp sugar. Leave in a warm location for 10 minutes and allow the yeast to bubble and rise.
In a bowl of a stand mixer combine flour, the rest of the sugar and salt. Add the risen milk and the water, and knead until it’s smooth and elastic, about 3 minutes or so.
Add the oil and knead in the stand mixer again for another 1-2 minutes.
Grease a glass bowl or metal mixing bowl. Put the dough in the bowl and cover with a dish towel. Put the bowl in a dark, warm location and allow it to rise for about 2 hours or until it doubles in size. (I put the dough in the Instant Pot on the YOGURT setting and covered it with a glass lid for one hour, which sped up the rising process). Your dough will be very sticky but this is normal.
While the dough is rising, use a medium bowl to combine the feta, mozzarella, and ricotta cheese. Add the salt, pepper, and chopped herbs (dill, chives, and parsley). Stir to combine and set aside.
Preheat the oven to 375 degrees.
Once the dough has risen, cut it into quarters. Working with each quarter individually, use a rolling pin to roll the dough out into a piece about 10” long and 6 inches wide. Roll the long edges in toward the center until a boat-shape is formed. Use your fingers to twist the corners and pinch them together.
Line a baking sheet with parchment paper and put the dough on the paper.
Divide the cheese mixture in quarters. Add ¼ of the cheese in each boat. Once all boats are filling, whisk up an egg in a small bowl. Use a pastry brush to brush the edges of the dough with the egg.
Put the tray in the oven and bake for about 15 minutes.
After 15 minutes, remove the sheet from the oven. Use a spoon to make a small pocket in the center of the cheese filling on each khachapuri. Crack one egg into each pocket. Then place a few dollops of butter (maybe ⅓ tsp each) around the cheese mixture and put the tray back in the oven. Bake until the eggs set (an additional 8-10 minutes). Enjoy!