Heat grill to medium-low for 10 to 15 minutes. Coat dough with 2 tablespoons olive oil, and stretch until it is roughly 14-by-12 inches. Place dough directly on grill grates, and cover grill. Cook, checking every 2 minutes, until dough bubbles, stiffens, and has visible grill marks on the bottom, about 4 minutes.
Flip dough. Cover grill, and cook 1 minute. Transfer dough from grill to a cutting board. Spread desired amount of heirloom sauce on cooked dough. Top with cheeses. Return pizza to grill, cover, and cook, checking after 2 minutes, until cheeses are melted and dough is fully cooked, about 3 minutes. Remove from grill and cool.
Dress arugula lightly with equal parts oil and vinegar, and season with salt and pepper. Top pizza with greens and serve.