Heat grill to medium-low.
Divide the dough into 2 equal pieces. On a lightly floured surface, roll each to a ¼-inch thickness. Brush each top with ½ tablespoon of the oil.
Transfer the dough to grill, oiled-side down. Cover grill and cook until the tops begin to bubble and the bottoms become crisp, about 3 minutes.
Working quickly, brush the top of each piece of dough with ½ tablespoon of the oil and turn over.
Divide the prosciutto evenly between the pizzas and continue to cook, uncovered, until the bottoms are golden and crisp, 2 to 3 minutes.
Remove from heat and layer with the tomato. Top with the arugula and drizzle with the remaining oil. Cut into pieces and serve on individual plates.