How to Make It
Prepare the dough. Combine the flour, yeast, salt, water, and butter in the bowl of a stand mixer fitted with a dough hook and knead for about 10 minutes to a soft and slightly sticky dough.
Lightly flour the dough in the bowl, cover with plastic wrap, and let rise for about 30 minutes, until doubled in volume.
Punch the dough down and divide it into 6 equal-size balls. Cover with a kitchen towel and let rise for another 15 minutes.
Preheat the oven to 400°F. Line two baking sheets with parchment.
Prepare the filling. Combine the cottage cheese, egg, mozzarella, and feta in a bowl.
Work with one piece of dough at a time. With a rolling pin, stretch and flatten the dough into an oval sheet ⅓-inch-thick. Place one-sixth of the filling in the center of the oval. Spread to within 1 inch of the edges. Roll the edges over to make a thick rim, pinching the narrow sides of the oval together to form a point at the ends. The pie should look like a kayak (see the photo above). Fill and shape the rest of the pies in the same manner.
Slide the pies onto the baking sheets and place on the bottom rack of the oven. Bake for 12 to 15 minutes, until the crust is golden and the bottom is firm and crusty. Remove the pies from the oven, leaving the oven on.
Prepare the topping. Break the eggs, one at a time, into a small bowl and carefully slide each into the pie. When all the pies are filled with the eggs, return the pan to the oven and bake for 5 minutes. Top each pie with a small cube of butter and serve immediately.