In a large mixing bowl, combine the mayonnaise, Greek yogurt, Dijon mustard, red wine vinegar, garlic powder, onion powder, oregano, salt, and black pepper. Mix well until smooth.
Add the shredded chicken to the bowl and stir until the chicken is fully coated with the dressing.
Gently fold in the diced tomatoes, cucumbers, red onion, black olives, mozzarella cheese, pepperoncini peppers, and chopped basil until evenly distributed.
Cover the bowl and refrigerate for at least 30 minutes to allow the flavors to meld.
Serve chilled on a bed of greens, in a sandwich, or in a wrap.