Lightly brush pizza dough with 2 tbsp olive oil. Press and roll out the dough into a long, flat shape about 1/2 inch thick, 5 inches wide and 14 inches long.
Place on the grill perpendicular to the grill grates. Grill on medium heat for 4-5 minutes, until the bottom is light golden brown and dark grill marks appear. Flip to the other side and grill for another 3-4 minutes until both sides are a light golden brown color.
Transfer to a sheet pan and spread the Stracchino cheese over the dough. Place the pan in the oven and broil for 4-5 minutes, until cheese has melted.
Top with the proscuitto and arugula. Sprinkle parmesan over it and lightly drizzle on the rest of the olive oil.