Generously brush one side of the pizza dough with oil; grill, oiled side down, until the underside is golden brown and the top begins to bubble, 3 to 5 minutes.
Quickly brush the top with oil; flip the crust.
Top with a thin layer of all four cheeses.
Grill until just melted and the crust is cooked through, 3 to 5 minutes more.
Slide the pizza with a large spatula onto a cutting board.