In a large bowl, combine the ground beef, salt, black pepper, cumin, smoked paprika, khmeli suneli, onion, bell pepper, garlic, parsley, cilantro, and shredded mozzarella. Mix gently until everything is evenly combined.
Lay a tortilla or lavash flat and spread a thin, even layer of the beef mixture over the entire surface without leaving a border. Roll tightly into a log, then slice into 1 to 1½-inch pinwheels.
Thread the pinwheels onto skewers if desired, or place them directly on a preheated medium grill. Cook for 15 to 20 minutes, turning every few minutes, until the tortillas are crisp and golden and the beef reaches an internal temperature of 165°F.
While the pinwheels cook, stir together all of the chimichurri ingredients and let the sauce rest for 10 to 15 minutes. Brush the chimichurri generously over the hot pinwheels as soon as they come off the grill, then serve immediately.