Bring a large pot of salted water to a rolling boil. Add the pasta and cook until just al dente according to package directions (usually 8–10 minutes). Drain and rinse briefly under cold water to stop the cooking, then toss the drained pasta with a teaspoon of olive oil to keep it from sticking. Set aside to cool completely.
While the pasta cooks, prepare the dressing: in a medium bowl whisk together the Greek yogurt, mayonnaise, basil pesto, lemon juice, minced garlic, honey, Parmesan, salt and pepper until smooth and evenly emulsified.
Taste the dressing and adjust seasoning — add a pinch more salt or another teaspoon of lemon juice if you want a brighter finish. Let the dressing sit for 5 minutes so the garlic and basil mellow and flavors meld.
In a large serving bowl combine the cooled pasta, chopped chicken, halved cherry tomatoes, diced red onion and most of the mozzarella (reserve a few pearls for garnish).
Pour about two-thirds of the dressing over the pasta mixture and toss gently with tongs until everything is evenly coated. Add the chopped greens and fold them in carefully — their texture will stay bright if added last. If the salad seems dry, stir in the remaining dressing a tablespoon at a time.
Taste and tweak: add salt, pepper, or a squeeze of lemon if needed. Transfer to a serving bowl, top with the reserved mozzarella pearls and a few whole basil leaves for color. Serve right away or refrigerate for up to 48 hours (see notes).