Preheat a grill to medium. Season the chicken with salt and pepper. On a lightly greased grill grate, cook the chicken through, turning once, about 10 minutes. Coarsely chop.
Meanwhile, in a pot of boiling, salted water, cook the pasta until al dente; drain and return to the pot. Add the chicken, vodka sauce and pecorino-romano; cover to keep warm.
Drizzle the EVOO on a baking sheet; turn the dough disks in the oil. Press each disk into a 6-inch round.
Place the dough on the grill, cover and cook until it bubbles on top and grill marks appear underneath, about 3 minutes. Flip the crusts and top with the pasta mixture, mozzarella and parsley. Cover and cook until the cheese is melted and the crusts are golden, about 3 minutes.