Divide pizza dough in half. Roll each half to a 10-inch circle. Brush with 1 tablespoon oil; set aside.
Heat remaining oil in a 12-inch cast iron skillet on a charcoal or gas grill directly over medium heat. Add prosciutto. Cover and cook for 2 minutes, stirring once. Add cherries and rosemary. Cover and cook for 2 minutes more or until softened, stirring once. Remove from heat. Stir in greens.
Place pizza dough on the grill rack. Cover and grill for 2 minutes. Turn dough over. Grill 2 minutes more. Top with cheese. Using a slotted spoon spoon cherry mixture over cheese. Grill, covered, for 1 to 2 minutes more or until cheese melts and crust is crisp.