Grate the English cucumber and place in a clean kitchen towel or cheesecloth; squeeze firmly to remove as much liquid as possible.
In a large mixing bowl, whisk together the Greek yogurt, minced garlic, lemon juice, olive oil, fresh dill, salt, and black pepper until smooth.
Fold the drained cucumber into the yogurt mixture until fully incorporated.
Add the shredded cooked chicken breast to the bowl and toss until the chicken is thoroughly coated in the tzatziki dressing.
Adjust seasoning with additional salt or lemon juice if necessary.
Cover and refrigerate for at least 30 minutes to allow the flavors to develop before serving.