For dough: In the bowl of a stand mixer, combine warm water, yeast, and sugar. Let sit for 5 min. to allow yeast to activate. Add flour, salt, and marjoram and mix on medium-low speed until a smooth, wet dough forms. Transfer dough to a clean greased bowl, cover, and let rise until doubled in size, about 1 hour. Once risen, turn dough out onto a lightly floured surface and divide into 4 equal portions. Place each piece of dough onto a piece of lightly floured parchment paper. Gently shape each portion into large ovals, about 8 inches wide and 10 inches long.
For filling: Place a pizza stone or heavy baking sheet into the oven and preheat to 500 degrees. Combine mozzarella, Muenster, feta, and marjoram. Pile a quarter of the mixture across each of the 4 ovals. Fold dough edges up over the cheese to form a border, about 2-inches wide all the way around. Twist ends once to form small “handles.” At this point, the uncooked khachapuri should resemble boats. Let them rise for 15 more min. Meanwhile, whisk 1 of the eggs in a small bowl. Trim the parchment paper to just larger than the khachapuri. Gently move the “boats” with parchment onto the pizza stone or baking sheet. You may have to bake the khachapuri in batches depending on the size of your stone. Brush the edges of each one with beaten egg and bake for 12 to 14 min., or until the dough is lightly golden. Crack 1 egg into the filling of each of the khachapuri and bake for 2 to 3 more min., just until the white of the egg is no longer translucent. Top each one with about 1/2 Tbsp. of cubed butter and a sprinkle of Urfa chile. Serve immediately.