Select a container that will fit the chicken and brine mixture but not be too big. Measure out the apple juice and water with a measuring cup and pour both into the container. Measure out the salt with a measuring cup and pour it into the juice mixture.
Stir the salt into the juice mixture with a whisk until it is completely dissolved (1 to 2 minutes), then put the chicken in. Make sure the chicken breasts are covered by the brine. Cover the container (with a lid if you have one, or plastic wrap) and put it in the fridge for 7.5 hours.
Remove the container from the fridge and dump out the brine mixture 30 minutes before you plan to put the chicken on the grill. This helps warm the chicken slightly so it will cook more evenly. Place the chicken on a cutting board and remove any excess liquid with a paper towel. Set aside.
Now is the time to get your Traeger (or other wood pellet grill or smoker) started up. Make sure there are pellets or wood in it and set it to 250 degrees F.
Measure out the seasonings with measuring spoons into a small bowl. Stir together with one of the measuring spoons then set aside briefly. Measure out the oil and pour over the chicken breasts, then rub it over the chicken with your hands until they are well covered.
Sprinkle half of the seasoning on one side of the breasts and rub it in, then flip them over and sprinkle the other half of the seasoning and rub that in. Make sure the chicken is well covered with seasoning.
Place your chicken on the Traeger using a pair of tongs or other utensil. If you have it, place a temperature probe into the thickest portion of one of the chicken breast (usually there is a thicker end). Set the probe to 160 degrees F. If you don’t have a temperature probe, check the chicken with a meat thermometer (we love our Thermapen) after 45 minutes to see where it’s at, then check again every 10 to 15 minutes and pull them once they reach between 160 and 165 degrees F.
Once the breasts are done smoking pull them off the Traeger and onto a clean cutting board. Let them rest for 5 to 10 minutes, then serve.