01 - Combine flour, instant yeast, sugar, and salt in a large bowl. Add warm water and olive oil, stirring until a shaggy dough forms. Knead on a floured surface for 8-10 minutes until smooth and elastic. Cover with plastic wrap and let rise in a warm, draft-free area for 1 hour until doubled in size.
02 - Preheat oven to 475°F. Line a large baking sheet with parchment paper and set aside.
03 - In a medium bowl, combine shredded mozzarella, crumbled feta, diced tomatoes, chopped parsley, dried oregano, and black pepper. Mix thoroughly and set aside until ready to use.
04 - Punch down the risen dough and divide into 4 equal portions. On a lightly floured surface, roll each piece into an oval shape measuring approximately 10 x 4 inches. Transfer to the prepared baking sheet.
05 - Spread the cheese and tomato mixture evenly over each dough oval, leaving a 1/2-inch border around the edges. Fold the edges over slightly and pinch the ends tightly to create the characteristic boat shape. Brush the dough edges with melted butter.
06 - Bake at 475°F for 10 minutes until the dough begins to set and the edges start turning golden brown. Remove from oven carefully.
07 - Crack one large egg into the center of each pide, being careful not to break the yolk. Return to the oven and bake for an additional 7-10 minutes until egg whites are fully set but yolks remain slightly runny, and the crust edges are golden brown and crispy.
08 - Remove from oven and brush the edges with additional melted butter if desired. Sprinkle with extra chopped parsley. Serve immediately while warm and eggs are runny.