Prepare the cucumber by grating it on the large holes of a box grater, then squeezing it very firmly in a clean towel to remove as much liquid as possible so the salad stays creamy.
In a medium bowl, whisk together the Greek yogurt, olive oil, lemon juice, dill, parsley, garlic, oregano, pepper, and salt until smooth and well combined.
Stir the well-drained grated cucumber into the yogurt mixture to create a thick tzatziki-style dressing, then taste and adjust the seasoning if needed.
Place the shredded rotisserie chicken into a large mixing bowl, breaking up any large clumps with your fingers so it is evenly sized.
Add the red onion and cherry tomatoes to the chicken, then pour the tzatziki dressing over the top and gently fold everything together until the chicken is well coated.
Fold in the crumbled feta if using, taste for seasoning, and add a pinch more salt or lemon juice to brighten the flavors if desired.
Cover the bowl and chill the tzatziki chicken salad for at least 20 to 30 minutes so the flavors can meld and the texture can set.
Serve scoops of the salad over baby spinach or mixed greens, or tuck it into pita or wraps, garnishing with extra dill and lemon wedges if you like.