Cook the tortellini: Bring a large pot of salted water to a boil. Add the tortellini and cook according to package directions until just tender. Do not overcook—you want them al dente so they hold up in the salad.
Rinse and cool: Drain the tortellini and rinse under cold water to stop the cooking.
Shake off excess water and spread on a sheet pan to cool completely.
Make the dressing: In a jar or bowl, whisk together olive oil, red wine vinegar, lemon juice, Dijon, garlic, oregano, dried basil, salt, pepper, and red pepper flakes. Taste and adjust seasoning.
Prep the mix-ins: While the pasta cools, chop the chicken, tomatoes, cucumber, bell pepper, and onion. Slice the olives and pepperoncini.
Tear or chop the herbs.
Toss the base: In a large bowl, combine the cooled tortellini, chicken, tomatoes, cucumber, bell pepper, red onion, olives, and pepperoncini.
Add cheese and herbs: Gently fold in the mozzarella pearls, Parmesan, parsley, and basil.
Dress it up: Pour about two-thirds of the dressing over the salad and toss gently to coat. Add more dressing if needed. The pasta will absorb some as it rests.
Rest and taste: Let the salad sit for 10–15 minutes (or chill up to a few hours).
Taste and adjust salt, pepper, or acid. A quick squeeze of lemon before serving brightens everything.
Serve: Enjoy chilled or at room temperature. Finish with a sprinkle of Parmesan and a drizzle of olive oil if you like.