Place chicken breasts and Italian dressing in a gallon-sized resealable bag. Let the chicken marinade in the fridge overnight.
Remove chicken from fridge and remove from marinade. Discard marinade.
Pat chicken breasts dry with a paper towel.
Rub all sides of chicken breasts with chicken rub.
Preheat smoker to 325 degrees and let the chicken sit covered in the rub while the smoker preheats so it can soak in the flavor of the rub.
Place chicken breasts on the smoker and close the lid on smoker. Let them cook for 15 minutes. Flip them over, close lid on smoker, and cook for 15 more minutes.
Check internal temperature of chicken breasts - if it reaches 165 degrees, you can pull the chicken breasts off the smoker. If not, continue smoking the chicken breasts for 5-10 more minutes or until it is fully cooked (it will depend on the thickness of your chicken breasts).
Remove chicken breasts, place on a plate, and cover with aluminum foil. Let chicken rest for 3-5 minutes before serving.
Notes
Try to get your chicken breasts to be all the same size before cooking them. I like to use the flat edge of a mallet to flatten the bigger chicken breasts so that they are uniform in size and cook in the same amount of time.
Some people like to baste their chicken, but we found that the marinade and rub leave it moist and juicy and it didn't even need any baste of glaze. But feel free to add a baste or glaze (even like BBQ) at the end of your cooking time.
This chicken is delicious as is, but is awesome to put on top of salads or use in any recipe that calls for cooked chicken! We love to make a lot of it and freeze it to use at a later time.