In a large pot, combine 2 quarts of water, salt, and sugar. Stir over medium heat until the salt and sugar dissolve completely.

Remove from heat, add the ice water and stir.

Submerge the turkey breast in the brine, making sure it's fully covered. Use a plate, a stack of plates or another heavy object to weigh it down if necessary.

Refrigerate for 8 to 24 hours.

Remove the turkey breast from the brine, and pat dry. Transfer the turkey to a large plate (or whatever you want) and let sit for 30 minutes to 1 hour.

In a small bowl, mix together all rub ingredients.

Preheat your smoker to 225 °F (107 °C) with the lid closed for about 15 minutes. Apple, cherry, pecan and maple are good wood choices.

Coat the turkey breast lightly with olive oil to help the rub stick, including between the skin and the meat. Generously apply the rub to the entire turkey breast, making sure to coat it evenly and get the area between the skin and meat.

Place the turkey breast on the smoker, skin side up, directly on the grill grates. Place the probe/themometer into the thickest part of the breast.

Smoke at 225 °F (107 °C) for about 2.5 to 4.5 hours (it needs about 35-45 minutes per pound), or until the internal temperature reaches 160 °F (71 °C) in the thickest part of the breast. If it’s starting to look dry, you can baste the turkey breast every hour with olive oil. If you want to crisp up the skin,

Once the turkey reaches the desired temperature, remove it from the smoker and let it rest for 15 to 20 minutes. During resting, the internal temperature should rise to 165 °F (74 °C). After the resting period, carve the turkey breast and serve.