Spray grill rack with nonstick non flammable cooking spray. Preheat both sides of grill to medium heat (300° to 350°). Turn off 1 side of grill.
Brush both sides of crust evenly with 1 tablespoon oil. Place crust on unlit side of grill. Grill, covered with grill lid, for 2 to 3 minutes on each side, or until lightly browned. Remove from grill.
Arrange chorizo and cheese on browned crust; return crust to unlit side of grill. Grill, covered with grill lid, for 4 minutes, or until cheese melts. Place pizza on a cutting board. Top with arugula; sprinkle with salt and pepper. Drizzle with remaining 2 tablespoons oil.