Heat grill to medium-high heat. Use tongs to coat the grill grates with an olive oil-soaked paper towel. Lightly flour the back of a sheet tray and stretch dough to desired size and shape. Brush pizza dough with olive oil so it doesn't stick when flipped on the grill. Slide dough off the sheet tray and onto the hot grill. Grill 4 to 6 minutes, or until golden brown on the bottom. Flip pizza using tongs. Add desired toppings (except garnishes). Close grill and continue cooking for another 4 to 6 minutes, or until toppings have melted and pizza is golden brown and crispy. Carefully remove from grill. Add garnishes, cut and enjoy!
Once pizza dough has charred on both sides, spread tomato sauce on crust. Sprinkle on a thin layer of shredded mozzarella. Place a few bell pepper rounds on crust. Crack an egg into each bell pepper round. Sprinkle with salt and pepper. Cover and cook for 4 to 6 minutes, or until the egg whites have set. Remove from grill and top with pieces of prosciutto and arugula. Cut and serve!