Preheat a grill or grill pan to medium. In a large bowl, combine the chicken with 1/2 cup parmigiano-reggiano, the bread crumbs, egg, tomato paste, two-thirds of the garlic, the parsley and basil; season with salt and pepper. Divide the mixture into 4 equal portions, then form each into 3 meatballs: Press 1 bocconcini into each meatball, then pinch the meat to seal. Coat the meatballs with 2 tablespoons EVOO, then thread onto skewers. Cover and grill, turning, until firm and cooked through, about 18 minutes.
In a large bowl, combine the lemon juice, anchovy paste, Worcestershire sauce, remaining garlic and 1/4 cup parmigiano-reggiano. Whisk in the remaining 6 tablespoons EVOO. Add the lettuce and radicchio and toss. Serve with the meatballs.