If using raw chicken, poach or bake until cooked through, then let cool. Shred or dice into bite-sized pieces.
In a large bowl, whisk together mayonnaise, Dijon mustard, lemon juice, salt, and pepper until smooth.
Add the chicken, celery, cucumber, fresh dill, and red onion (if using) to the bowl.
Gently fold everything together until evenly coated with the dressing.
Taste and adjust seasoning with more salt, pepper, or lemon juice as needed.
Cover and refrigerate for at least 30 minutes (or up to overnight) to let flavors meld.
Serve cold on bread, over greens, or with crackers.
