In a medium bowl, combine mayonnaise (and Greek yogurt, if using), Dijon mustard, lemon juice, salt, and pepper.
Fold in the cooked chicken, dried cranberries, celery, and green onion until evenly coated.
Taste and adjust seasoning—add more salt, pepper, or lemon juice as desired.
Arrange the apple slices on a platter or individual plates, overlapping slightly.
Top each slice (or clusters of slices) with spoonfuls of the cranberry chicken salad.
Garnish with chopped parsley or walnuts, if using.
Serve immediately for best texture—if prepping ahead, toss apple slices with a bit of lemon juice to prevent browning and store chicken salad separately until ready to assemble.