Place a rack in top third of oven; preheat to 550? (or prepare a grill for high heat; preheat 5 minutes until 500?). Place a pizza stone in oven to preheat.

Weigh out 4 (8-ounce) balls of dough. On a lightly floured surface, roll each ball to an 8"-diameter round. Lightly dust top of dough with flour. Fold top half of dough over the bottom, but do not press to seal. Carefully place on preheated pizza stone.

Bake crust until puffed, golden brown, and some dark spots appear, about 10 minutes. Let cool slightly.

Carefully open the seam where you folded the dough in half. Spread 3 tablespoons jam on the bottom. Top with 4 ounces torn burrata, 1/4 cup arugula, and 3 pieces prosciutto. Drizzle with oil; season with salt and pepper. Repeat with remaining dough, jam, burrata, arugula, prosciutto, oil, salt, and pepper for a total of 4 sandwiches.