Grate the English cucumber and use a clean kitchen towel or cheesecloth to squeeze out as much moisture as possible.
In a large mixing bowl, combine the Greek yogurt, minced garlic, lemon juice, olive oil, chopped dill, salt, and black pepper.
Stir the drained cucumber and diced red onion into the yogurt base until well combined.
Add the shredded chicken to the yogurt mixture and fold gently until all chicken pieces are thoroughly coated.
Cover and refrigerate for a minimum of 30 minutes to allow the flavors to develop before serving.