Heat a grill pan over medium-high heat or preheat a gas or charcoal grill.
Brush each side of the pita breads with ½ teaspoon olive oil and grill for 2 to 3 minutes on each side, until crisp.
Remove from the grill and cool slightly.
In a large skillet, heat 1 tablespoon olive oil over medium-high heat.
Crack the eggs directly into the pan and cook until the egg whites are set, 2 to 3 minutes.
Combine the mascarpone cheese, lemon zest, ½ teaspoon salt, and ½ teaspoon pepper in a small bowl.
In a medium bowl, whisk together 3 tablespoons olive oil, the lemon juice, 1 teaspoon salt, and ½ teaspoon pepper until smooth. Add the arugula and toss until coated. Spread each pita with 2 tablespoons of the mascarpone mixture.
Divide the prosciutto on top. Divide the arugula and mound on top of the prosciutto.
Carefully place a fried egg on top of each pita. Season the eggs with a pinch of salt and pepper, and serve.