In a medium bowl, combine the dressing ingredients: mayonnaise, grated Parmesan, red wine vinegar, banana pepper juice, oregano, and garlic powder. Whisk until smooth and set aside.

In a large mixing bowl, add the shredded rotisserie chicken, diced deli meats, and diced red onion. Toss in the halved grape tomatoes, sliced banana peppers, cubed provolone cheese, and crumbled feta cheese. Mix well.

Pour about three-quarters of the dressing over the chicken and vegetable mixture. Toss carefully until all ingredients are evenly coated. Add more dressing as preferred.

Cover the bowl and refrigerate for at least 30 minutes before serving to meld flavors. Serve on toasted croissants, with crackers, or as a standalone dish.