Prepare a gas grill for medium heat or prepare a charcoal grill for medium indirect heat.
Rub the dough with the oil. On a clean work surface, pat and stretch the dough into a rustic 12-inch circle (no need for perfect shaping). Carefully transfer the dough to the grill grates; cook until grill marks appear and the crust is golden, about 6 minutes. Flip, spread the sauce on the crust in an even layer and scatter with the mozzarella. Top a quarter of the pizza with the prosciutto, a quarter with the artichokes, a quarter with the mushrooms and a quarter with the olives. Cover and grill until the cheese is melted, the crust is cooked through and the underside is charred in spots, about 6 minutes more. Transfer to a cutting board; let stand about 5 minutes. Top with the basil, if using, and cut into slices.
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