Make dough: Combine milk, water, yeast, and sugar in a large bowl. Let stand until yeast forms a creamy foam, about 5 minutes. Add 2 cups flour, olive oil, and salt; mix until a sticky dough forms.

Turn dough out onto a floured surface. Knead for about 3 minutes, adding in remaining 1/4 cup 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.

Preheat the oven to 475 degrees F (245 degrees C).

While dough rises, prepare cheese blend: Toss together feta, Monterey Jack, and mozzarella cheese in a medium bowl. Refrigerate cheese mixture until needed.

Punch down dough and turn out onto a floured surface. Flatten slightly and cut in half. Place 1 dough half on a sheet of parchment paper dusted with flour. Roll out into a rectangle about 1/8- to 1/4-inch thick.

Arrange 1/4 of the cheese mixture in 2 rows, 1 to 2 inches from the longest sides. Roll sides up tightly over cheese, forming a canoe shape. Pinch and twist ends to seal. Trim excess parchment and lift dough onto a baking sheet. Shape and stuff remaining dough half. Fill canoes with 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 the oven and bake until eggs are mostly set, 3 to 4 minutes. Place 2 pats butter over each egg, then sprinkle with salt and cayenne. Plate khachapuri; tear off one end and use it to stir butter into egg.