Begin by preparing your chicken, take a mixing bowl, and add brown sugar and salt. Mix to combine. Add your chicken breasts, then cover in water. Refrigerate and allow your brine to soak into your chicken and marinade for 1-2 hours, but no more than 6 hours.
Prepare the grill for smoking by preheating your smoker to 225°F.
In a small mixing bowl, combine your chicken rub seasonings. Remove your chicken from the brine, and pat dry. Brush with Dijon mustard, then dip into the seasoning mixture on both sides to apply the rub to the chicken. Now your chicken breasts are ready to smoke.
Place your chicken breasts onto your smoker and insert a digital read temperature probe into the largest chicken breast so that you can keep an eye on the internal temperature of your chicken. Smoke your chicken breasts until they reach the internal temperature of 160°F. Remove from the grill.
Tent your chicken with a sheet of aluminum foil, while you allow your chicken to rest for 5-10 minutes. Then serve and enjoy!
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