In a large bowl, combine the chicken, tomatoes, celery, and onion. Add the peanut sauce, mayonnaise, and lime juice and toss to combine. Season with salt and pepper. (At this point, you can refrigerate the salad for up to 3 days.)
Place the lettuce on 4 individual plates and top with the chicken mixture. Sprinkle the peanuts over the top just before serving.
Thai Peanut Sauce:
In a medium saucepan, whisk together the sauce ingredients until well blended and set the pan over medium heat. Bring to a simmer and let simmer for 10 minutes.
The leftover peanut sauce will keep in the refrigerator up to 4 days or in the freezer up to 3 months; thaw completely in the refrigerator or microwave for about 3 minutes on LOW before using.