Heat grill to medium-low. Divide the dough into 2 equal pieces. On a lightly floured surface, roll each to a 1/4-inch thickness. Brush each top with 1/2 tablespoon of the oil. Transfer the dough to the 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 1/2 tablespoon of the oil and turn over. Divide the prosciutto evenly among them 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 divide among individual plates.
Tip: To make this recipe kid-friendly, replace the prosciutto with sliced ham and omit the arugula.