Prepare the chicken by boiling or baking until fully cooked, then shred into bite-sized pieces.
In a large mixing bowl, combine Greek yogurt, diced cucumber, minced garlic, lemon juice, olive oil, and dried oregano. Mix until smooth and season with salt and pepper.
Add shredded chicken, cherry tomatoes, crumbled feta cheese, chopped red onion, and fresh dill to the tzatziki sauce. If using, add rice or quinoa. Gently fold until evenly coated.
Preheat oven to 350°F. Transfer mixture to a greased 9x13-inch baking dish, sprinkle breadcrumbs on top if desired, and bake for 25-30 minutes until heated through and top is golden.
Let cool for 5 minutes before serving.