Combine milk, water, sugar, and yeast in a large bowl. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes. Add olive oil, 2 cups flour, and salt. Mix until a sticky dough comes together.
Turn dough out onto a floured surface. Knead for about 3 minutes, adding in remaining flour as needed, until dough is very soft but still workable. Form into a ball and place in a lightly oiled bowl. Cover and let rise in a warm spot until doubled in volume, 1 to 1 1/2 hours.
Punch down dough and turn out onto a floured surface. Flatten slightly and cut in half. Dust a large sheet of parchment paper with flour. Place 1 dough half on top and roll out into a square about 1/8- to 1/4-inch thick, stretching out the corners as needed. Arrange 1/4 of the cheese in 2 rows, 1 to 2 inches from the longest sides. Roll sides up tightly over the cheese, forming a canoe shape around the dough. Pinch and twist ends to seal.
Cut off excess parchment around the dough; lift dough onto a baking sheet. Repeat shaping and stuffing the other dough half. Fill the canoe with the remaining cheese mixture.
Bake in the preheated oven for 15 minutes. Form a well in the center of each khachapuri using a spoon; crack an egg inside each well.
Return to oven and bake until eggs are mostly set, 3 to 4 minutes. Place 2 pats of Andrew & Everett Butter over each filling and sprinkle salt and cayenne on top. Plate khachapuri; tear off one end and use it to stir the butter into the egg.