Fill a 2 1/2 quart pot with water and add a little salt to flavor the water. When the water is boiling, add the chicken and cook until tender and no longer pink in the center, about 30 minutes. Remove the chicken and allow it to cool enough to handle.
In the mean time, add the celery and onion to the food processor and pulse for about 30 seconds until the vegetables are finely chopped. Add them to a bowl and chop the pecans if you didn't buy them already chopped. You can do that by hand or put them in the food processor--just don't chop them too small. You want them to still have some crunch.
Once the chicken is cool enough to handle but still warm, remove any tough areas from the chicken and break the meat into quarter-size chunks and throw it in the food processor. I do my chicken breasts one at a time because I have a small food processor. If you have a large one, you can do them all at once. Pulse the machine until the meat is finely chopped and no large chunks remain. This usually takes about 20 seconds or so. Add your chicken to the vegetables and pecans.
Stir in the mayonnaise, salt, pepper and lemon. Taste for seasoning. Wrap the salad in plastic wrap and allow it to sit in the fridge several hours or overnight for the best flavor.