01 - Combine flour, instant yeast, sugar, and salt in a large mixing bowl. Add warm water and olive oil, stirring until a shaggy dough forms. Knead on a lightly floured surface for 8 to 10 minutes until smooth and elastic. Cover with plastic wrap and let rise in a warm, draft-free area for 1 hour or until doubled in volume.
02 - Preheat oven to 475°F. Line a baking sheet with parchment paper. Meanwhile, mix shredded mozzarella, crumbled feta, diced tomatoes, chopped parsley, dried oregano, and black pepper in a medium bowl until well combined.
03 - Punch down the risen dough and divide into 4 equal portions. On a lightly floured surface, roll each piece into an oval approximately 10 x 4 inches. Transfer to the prepared baking sheet.
04 - Distribute the cheese and tomato mixture evenly over each dough oval, leaving a ½-inch border. Fold the edges over slightly and pinch the ends to create the classic boat shape. Brush the dough edges with melted butter.
05 - Bake for 10 minutes until the edges begin to turn golden and the cheese starts to melt.
06 - Remove from oven and carefully crack one egg into the center of each pide. Return to oven and bake for 7 to 10 minutes until egg whites are set but yolks remain slightly runny, and the crust is golden brown.
07 - Remove from oven and brush edges with additional melted butter if desired. Garnish with extra chopped parsley. Serve immediately while warm.