Whisk together Greek yogurt, mayonnaise, vinegar, salt, pepper, garlic powder, dill and dijon mustard in a bowl.
Fold in shallots and shredded chicken. Mix well. Cover and refrigerate for at least 1 hour to let the flavors meld.
To make the sandwiches, spoon chicken salad onto one slice of bread. Top with lettuce and another slice of bread. Repeat with remaining ingredients to make 4 sandwiches. Serve.