Heat a grill pan over medium-high heat.
Brush each side of the pita breads with 1/2 teaspoon olive oil and grill 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, 1/2 teaspoon salt and 1/2 teaspoon pepper in a small bowl. In a medium bowl, whisk together 3 tablespoons olive oil, the lemon juice, 1 teaspoon salt and 1/2 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. Serve.