Remove the dough from the refrigerator, if necessary, about 1 hour before grilling so that the dough is easier to shape.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and preheat a pizza stone for at least 15 minutes, following manufacturer's instructions.
In a small bowl whisk 3 tablespoons oil, the garlic, and salt.
Using a rolling pin on a lightly floured work surface, roll out each ball of dough, one ball at a time, into a round about 12 inches wide and ¼ inch thick. (If the dough retracts, cover it with a kitchen towel, let it rest for 5 minutes, and then continue.) Set the first round aside while you roll out the second.
Carefully transfer the first round of pizza dough onto a pizza peel (or rimless baking sheet) lightly coated with flour. Brush the dough with half of the garlic oil. Sprinkle ¼ teaspoon red pepper flakes evenly over the dough. Arrange half of the mozzarella slices over the dough in single layer. Tear the prosciutto into pieces and lay half of them over and around the mozzarella. Sprinkle with half (⅓ cup) of the grated cheese.
Slide the first pizza onto the preheated pizza stone. Cook over direct medium heat, with the lid closed, until the crust is golden brown and the cheese is melted, 10 to 12 minutes.
Using a pizza peel or a large spatula, transfer the pizza from the stone to a large cutting board or baking sheet and immediately scatter half of the arugula over the pizza. Let the pizza rest for a few minutes. Before serving, lightly drizzle the pizza with olive oil and sprinkle with black pepper. Cut into wedges and serve warm.
Repeat the steps with the remaining dough round and topping ingredients.