Shred the chicken apart until you have 2 cups of chicken shredded into fairly large chunks. Dice the avocados into medium-sized pieces, mix with 1 T of the lime juice, and season avocado with salt to taste. Thinly slice the green onion and finely chop the cilantro. Mix mayo and 1 T lime juice to make the dressing.
Put the chicken into a bowl large enough to hold all the salad ingredients. Add the sliced green onions and dressing and toss until all the chicken is coated with dressing. Add the avocado and any lime juice in the bottom of the bowl and gently combine with the chicken. Then add the chopped cilantro and gently mix into the salad, just until it is barely combined.
Serve right away or chill for a while before serving. This could be served inside pita bread or sandwich bread, or inside crisp lettuce cups, but we just ate it as a salad. (I would imagine this would be fine in the fridge overnight but I didn't have any left over when I made it!).