Wash and dry the cucumbers. Using a vegetable peeler or mandoline, slice the cucumbers lengthwise into thin, wide strips about 1/8 inch thick.
In a medium bowl, combine the shredded chicken breast, Greek yogurt, mayonnaise, Dijon mustard, chopped celery, red onion, dill, parsley, lemon juice, and garlic powder.
Mix all ingredients thoroughly until well combined. Season with salt and black pepper to taste.
Lay a cucumber strip flat on a clean surface. Place about 2 tablespoons of the chicken salad mixture at one end of the strip.
Carefully roll the cucumber strip around the chicken salad filling to form a tight roll.
Repeat the rolling process with the remaining cucumber strips and chicken salad mixture.
Arrange the cucumber rolls on a serving plate and serve immediately or chill in the refrigerator for up to 1 hour before serving.