Pat the chicken breasts dry with paper towels and, if using, lightly brush both sides with olive oil to help the rub adhere.
In a small bowl, combine light brown sugar, smoked paprika, garlic powder, onion powder, kosher salt, and black pepper; stir until evenly mixed.
Generously sprinkle the rub over both sides of the chicken breasts and gently press it into the meat; for deeper flavor, refrigerate for 30–60 minutes (optional).
Preheat your smoker to 225°F and add apple or cherry wood chips or pellets.
Place the seasoned chicken breasts directly on the smoker grates and smoke until an instant-read thermometer inserted into the thickest part registers 165°F, about 1.5–2.5 hours.
Remove the chicken from the smoker, tent loosely with foil, and let rest for 5–10 minutes.
Slice or shred the smoked chicken and serve with your favorite sides or use in salads and sandwiches.