Heat a grill pan (or just use the skillet you'll be frying the egg in) over medium high heat. Brush the pita with some olive oil and cook for 2 to 3 minutes each side until browning and as crispy as you like. Remove and let cool slightly.
Combine the cheese with the zest from half the lemon, ½ teaspoon salt, and ½ teaspoon pepper in a small bowl.
In a bowl, whisk together 3 tablespoons olive oil, the juice from the lemon (3 tablespoons), 1 teaspoon salt, and ½ teaspoon pepper, until smooth. Add the arugula and toss to coat.
Heat 1 tablespoon of the oil, in a large skillet over medium-high heat. Crack the eggs into the pan and cook until your desired level of doneness. 2-3 minutes to get the whites to set, but leaves the yolk runny.
Spread each pita with 2 tablespoons of the mascarpone lemon mixture. Divide the prosciutto evenly and then divide the arugula and mound it on top of the prosciutto. Top with the fried egg and sprinkle with salt and pepper.
Notes
You can substitute bacon, pancetta, or thin sliced ham for the prosciutto.You can substitute cream cheese, Greek yogurt, or ricotta cheese for the mascarpone.Greens - baby kale, baby spinach, baby arugula, or microgreens all work.