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 the dough for 8 to 10 minutes until smooth and elastic. Cover and let rise in a warm location for 1 hour or until doubled in size.
02 - Set oven temperature to 475°F. Line a baking sheet with parchment paper.
03 - Mix shredded mozzarella, crumbled feta, diced tomatoes, chopped parsley, dried oregano, and black pepper in a bowl until well combined.
04 - Punch down the risen dough and divide into 4 equal portions. Roll each piece into an oval shape measuring approximately 10 x 4 inches.
05 - Place dough ovals on the prepared baking sheet. Distribute the cheese and tomato mixture evenly across each oval, leaving a ½-inch border. Fold the edges inward and pinch the ends to create the traditional boat shape.
06 - Brush the dough edges with melted butter. Bake for 10 minutes until lightly golden.
07 - Remove from oven and carefully crack one egg into the center of each pide. Return to oven and bake for 7 to 10 minutes longer, until egg whites are set but yolks remain slightly runny and crust is golden brown.
08 - Remove from oven, brush edges with additional melted butter if desired, and sprinkle with fresh parsley. Serve immediately while warm.