There's only so much we can say about this recipe. You could combine eggs, cheese, and bread in 1,000 different ways, and we'd love them all... but we'd never love them like this.
We discovered this is actually a traditional street food in Armenia (our great-grandfathers name is actually Katchperooni, go figure), though we had never seen or heard of it until we came across this Savuer post.
Sometimes we make a recipe and it really falls short of our expectations, or sometimes we all have different reactions to the same thing we made. But this, we were losing our ever-loving minds over this. The feta/Muenster cheese combo is salty and gooey, the dough is pillowly and perfect to dunk in that runny-egged perfection center, and basically that day we decided, we'd devote our lives to making Khachpuri for everyone we know if it were an option. I won't try to tell you that Kate and I shared these. They were both gone in 20 minutes.
Yield - well, shamelessly served 2, probably could actually serve 4-6
1 tsp. active dry yeast
1⁄4 tsp. sugar
1 tbsp. olive oil, plus more for greasing
1 1⁄4 cups flour, plus more for dusting
1 tsp. kosher salt
2 cups shredded Muenster cheese
1 1/2 cup crumbled feta cheese
2 tbsp. unsalted butter, cubed
Combine yeast, sugar, and 2⁄3 cup luke warm water in bowl; let stand until foamy, about 10 minutes. Add oil, flour, and salt; mix with a wooden spoon until a soft dough forms. Transfer to a lightly floured surface and knead until smooth and elastic, 4 minutes. Transfer to a lightly oiled bowl and cover loosely with plastic wrap. Set in a warm place until doubled in size, about 45 minutes.
Place a pizza stone on a rack in lower third of oven. Heat oven to 500° for 1 hour. Combine cheeses and set aside. Punch down dough and divide in half. On a piece of lightly floured parchment paper, roll half of dough into a 10" circle about 1⁄8" thick. On one side of the circle, tightly roll dough about a third of the way toward the center. Repeat on the opposite end. There should be a 2"-3" space between the two rolls. Pinch the open ends of the rolls together and twist to seal, making a boat shape; place half the cheese mixture in the middle; repeat with remaining dough and cheese. Transfer boats on paper to stone; bake until golden brown, 14–16 minutes. Crack 1 egg into the center of each boat. Return to oven until egg white is slightly set, 3–4 minutes. Place 1 tbsp. butter on each bread; serve hot.