In a large mixing bowl, whisk together the avocado oil mayonnaise, Dijon mustard, lemon juice, garlic powder, salt, and pepper until the emulsion is smooth.
Fold in the minced fresh dill, diced celery, and chopped green onions into the dressing base.
Add the shredded cooked chicken to the bowl.
Using a spatula or large spoon, toss the chicken thoroughly until every piece is evenly coated with the dressing.
Taste and adjust salt or pepper if necessary.
Serve immediately inside lettuce cups or refrigerate in an airtight container for up to 5 days.