Heat grill to medium. Divide pizza dough in half and shape each piece into a 5- by 14-inch rectangle.
Toss peaches with oil and ¼ teaspoon each salt and pepper. Grill dough and peaches until lightly charred, 3 to 5 minutes per side. Transfer peaches to large bowl and pizza dough to cutting board.
Toss peaches with lemon juice, tarragon, and pinch each salt and pepper. Sprinkle Gruyère over dough and grill, covered, until melted, 2 to 3 minutes.
Add arugula to peaches and toss to coat. Top cheesy pizza dough with prosciutto and peach-arugula salad.