Warm the water until it is warm but not too hot to touch. In a large bowl, add the active dry yeast, sugar, and warm water. Wait about five minutes for yeast to double in size.
Once the yeast has foamed and doubled in size, add the salt, flour, and oil. Turn the mixer on low speed until combined. Higher the speed and let knead until the dough comes together.
Remove the dough from the mixer and form a ball using excess flour. Place the dough back in the ball and cover with a clean kitchen towel. Place in a warm area for about 1 hour to rise.
Once the dough has doubled in size, form into dough balls using excess flour. You should get about one dozen large dough balls or, 2 dozen smaller ones. Cover with kitchen towel for another 30 min to rise again.
While the dough is rising, prepare the filling. Shred the block of mozzarella, and feta cheese. Combine 3 cups of mozzarella with 1 ½ cups of feta. Finely chop some parsley and add it in along with black pepper. Mix well to combine and set aside.
Preheat the oven to 425F and place oven rack in the middle. Prepare a baking tray with parchment paper.
Roll out the dough to a 6-inch oval. Place about 3-4 tablespoons of filling in the middle. Bring 2 sides together on each end and pinch to close.
Whisk 1 egg well. Using a pastry brush, brush all the fatayer on the sheet right before putting in the oven. Bake for 12-14 minutes until golden brown and cheese is melted.