If using raw chicken: place 2 boneless chicken breasts in a single layer in a saucepan and cover with water (or low-sodium chicken broth). Bring to a gentle simmer over medium heat, reduce to low and simmer until the chicken reaches 165°F, 12–15 minutes depending on thickness. Remove and let rest 5 minutes, then shred with two forks. Alternatively, use leftover or rotisserie chicken and shred.
While the chicken cooks, whisk together the mayonnaise, lime juice, salt, and pepper in a medium bowl until smooth; taste and adjust seasoning. This is your creamy dressing.
Dice the avocado and toss it immediately with a teaspoon of the lime juice from the dressing (reserve the rest) to slow browning and add brightness. Set aside a few small avocado pieces to use as a garnish if you like.
In a large mixing bowl combine the shredded chicken, chopped celery, diced red onion, and chopped cilantro. Gently fold in the avocado pieces so they don't turn to mush.
Pour the dressing over the chicken mixture and use a spatula to fold everything together until evenly coated; press a few chunks of avocado against the spatula to lightly mash into the dressing for extra creaminess if you prefer a smoother texture.
Cover and chill for at least 10 minutes to let the flavors meld (or serve immediately if short on time). Taste again and adjust salt, pepper, or lime to balance the flavor before serving.
Serve the salad on toasted bread or in lettuce leaves, or enjoy it straight from the bowl. Garnish with reserved avocado pieces and an extra cilantro sprinkle if desired.