Cook the pasta al dente. Boil in well-salted water according to package directions minus 1 minute. You want a firm bite so it holds up in the dressing.
Rinse and cool. Drain the pasta and rinse briefly under cool water to stop the cooking. Shake off excess water well so the dressing doesn’t get diluted.
Whisk the dressing. In a bowl or jar, combine olive oil, red wine vinegar, lemon juice, Dijon, garlic, oregano, dried basil, salt, pepper, and red pepper flakes.
Whisk or shake until emulsified.
Prep the mix-ins. Chop tomatoes, cucumber, red onion, bell pepper, olives, and pepperoncini. Dice or shred the cooked chicken. Rinse and drain chickpeas.
Chop parsley and basil.
Toss while pasta is slightly warm. In a large bowl, combine pasta, chicken, chickpeas, veggies, and herbs. Pour in about two-thirds of the dressing and toss gently to coat.
Add cheese and Parmesan. Fold in mozzarella and grated Parmesan. Taste and add more dressing, salt, or pepper as needed.
Rest for better flavor. Let the salad sit for 15–20 minutes (or chill for up to 2 hours) to allow the flavors to meld.
Stir again before serving and add a splash of dressing if it looks dry.
Finish and serve. Top with a little extra Parmesan and fresh basil. Serve cold or at cool room temperature.