1. 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.
2. 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.
3. Preheat the oven to 475 degrees F (245 degrees C). Combine Monterey Jack, mozzarella, and feta cheeses in a bowl. Toss together and refrigerate until needed.
4. 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.
5. 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.
6. 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.
7. Return to oven and bake until eggs are mostly set, 3 to 4 minutes. Place 2 pats of 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.
Footnotes:
With a cheese-stuffed crust and an egg baked into the center, this boat-shaped Georgian bread is so fun and versatile, there's only a matter of time before it's delivered to every home for pizza night and ordered from every brunch menu the morning after. They take a little finesse but the payoff is well worth it; after all, what more could you want other than cheesy, soft-while-chewy bread dipped in its own bed of runny egg yolk and molten cheese?
