Shred the cooked chicken into small, uniform pieces and place in a large bowl.
In a separate medium bowl, whisk together mayonnaise, sweet pickle relish, sugar, apple cider vinegar, yellow mustard, salt, black pepper, and celery seed until smooth and well combined.
Pour the dressing over the shredded chicken and stir until every piece is evenly coated.
Taste and adjust seasoning—add more sugar for sweetness or more vinegar for tang.
Cover the bowl with plastic wrap or transfer to an airtight container and refrigerate for at least 30 minutes to allow flavors to meld.
Serve chilled on bread, croissants, lettuce cups, or with crackers.