Place a rack in the top third of the oven; preheat to 550° (or prepare a grill for high heat; preheat 5 minutes until 500°). Place a pizza stone in the 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 the top of the dough with flour. Fold the top half of the dough over the bottom, but do not press to seal. Carefully place on the preheated pizza stone.

Bake the 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 of jam on the bottom. Top with 4 ounces of torn burrata, 1/4 cup of arugula, and 3 pieces of prosciutto. Drizzle with oil; season with salt and pepper. Repeat with the remaining dough, jam, burrata, arugula, prosciutto, oil, salt, and pepper for a total of 4 sandwiches.