To a jar with a tight-fitting lid, add the Parmesan, olive oil, mustard, vinegar and salt. Screw on the top and shake vigorously to emulsify. Set aside to let the flavors marry.
In a large bowl, toss together the romaine, radicchio, basil, cucumber, avocado and pita chips. Pour the dressing over the salad, sprinkle with the salt and toss gently to coat and combine.