Combine yeast, sugar, and ⅔ cup warm water in a medium bowl. Let stand until foamy, about 10 minutes. If it isn’t foamy, try again! Your yeast is dead or water was not warm.
Then add 1 tablespoon olive oil, flour, and 1 teaspoon salt and mix until a soft dough forms. Knead on a floured surface until the dough is smooth and elastic, about four minutes. Then transfer to a lightly greased bowl, cover, and set in a warm place for an hour or until doubled in size.
While your dough is rising, make the cheesy filling. Head 1 tablespoon olive oil in a large sauté pan. Sauté kale with a pinch of salt until softened, cool, and then add ricotta cheese, munster cheese and remaining salt to taste. Combine well.
When ready to bake, preheat the oven to 450 degrees F. Punch down the dough and divide in two. Roll out on piece of dough on a piece of parchment paper (or silpat) into a 1/8 inch thick oval. Spread half of the cheese mixture on the dough and leave a 1/2 inch boarder around the edge.
On one side of the oval, roll the dough up towards the cheese. Then do the same with the other side. Pinch the open ends together and twist to secure into a boat shape. Repeat with the other half.
Transfer boats on paper to a pizza stone or baking sheet. Bake 14 minutes until golden brown. Then crack 1 egg onto the center of each piece. Return the khachapuri to the oven and bake another 3-4 minutes until the egg white is slightly set. Place butter on bread if desired and serve!