1) Combine vegetable stock with herbs in a food processor. Turn processor on and stream in the extra-virgin olive oil.
2) 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.
3) 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.
4) Fill sub rolls with the chicken salad and serve with a few olive oil crisps alongside.