Combine vegetable stock with herbs in food processor. Turn processor on and stream in the extra-virgin olive oil. Finely chop the garlic then mash into a paste with some salt using the flat side of your knife. Transfer basil dressing to a salad bowl and stir in the garlic paste along with the Parmigiano-Reggiano. Add the tomatoes, and onions. Remove the skin from the chicken and chop or shred into bite-size pieces to yield about 4 cups. Add the chicken to the salad bowl and season with black pepper. Toss the salad and adjust seasonings. Fill sub rolls with the chicken salad and serve with a few olive oil chips alongside.