Cut both avocados in half and carefully remove the pits.
Scoop the flesh from one avocado into a large mixing bowl. Mash it with a fork until mostly smooth.
Add the mayonnaise, finely chopped celery, onion, cilantro, lime juice, and salt.
Stir and mash everything together until the avocado and mayonnaise form a thick, creamy dressing with the vegetables evenly distributed.
Add the shredded cooked chicken and gently stir until every piece is coated in the avocado mixture.
Cut the remaining avocado into approximately 1/4- to 1/3-inch pieces.
Gently fold the avocado pieces into the Creamy Avocado Chicken Salad, taking care not to mash them completely. Keeping some avocado chunky creates a more interesting texture.
Taste the salad and season with freshly ground black pepper. Add additional salt or lime juice as desired.
Serve immediately with toasted bread, crackers, tortilla chips, or your favorite fresh vegetables.