Preheat the smoker to 225°F.
While the smoker is preheating, prepare the rub. Combine 1 Tablespoon light brown sugar, 1 Tablespoon smoked paprika, 1 Tablespoon coarse salt, ½ Tablespoon garlic powder, ½ Tablespoon dried oregano, 1 teaspoon onion powder, ½ teaspoon dried thyme, ½ teaspoon ground black pepper, ½ teaspoon white pepper, ½ teaspoon cajun seasoning in a medium bowl. Stir to combine. Set aside.
Optional Step: To turn this into a wet rub, combine 2 Tablespoons of olive in with the dry rub. Mix well and spread evenly over each piece of chicken.
Using paper towels, pat dry each piece of chicken to remove any excess moisture.
Next, generously cover each breast with the dry rub while using a bit of pressure to press the spices into the meat. Make sure the whole piece is completely covered.
When the smoker has preheated, put the chicken breasts directly on the grates.
To monitor the internal temperature, insert a digital meat thermometer at a slight downward angle in the center of the breast.
When the internal temperature of the chicken reaches 165°F (about 1.5-2 hours later), remove the chicken breasts from the smoker and allow them to rest for 3-5 minutes.
Using a sharp knife, slice the smoked chicken breast into strips and serve immediately.