Place chicken in a large bowl or plastic freezer bag and coat with about ½ cup of caesar dressing.
Preheat oven to 425 degrees.
Remove excess dressing from chicken and place in a single layer on a parchment lined baking sheet. Season with salt and pepper and bake until evenly browned and cooked through, about 12-15 minutes for small boneless, skinless chicken tenderloins, longer for chicken breasts or thighs.
Turn off oven and remove chicken. Place chicken to the side and tent with foil to keep warm.
Lightly toast the pitas in a single layer in the still warm oven.
Spread 2 tablespoons caesar dressing in the middle of each toasted pitas.
Divide arugula, celery, cucumber, red onion, chicken and cashews evenly among wraps.
Drizzle another tablespoon dressing over top and fold in half.
Wrap bottom with parchment paper and secure with twine or a pick.
Serve immediately with a side of caesar dressing for dipping.