Pat the chicken dry with a paper towel. Using a pastry brush, apply olive oil to all sides of each breast.
In a small bowl, whisk together the brown sugar and seasonings. Use the pastry brush to liberally apply the mixture on all sides of the chicken breasts, then transfer them to a rimmed sheet pan or plate.
Using tongs, place chicken breasts directly on the grill plate of the smoker.
Smoke for about 1 hour, or until the internal temperature at the thickest portion of the breast reaches 162°F. Remove from the smoker and allow meat to rest 15 minutes before slicing. *During resting time, the chicken will continue to cook to a safe internal temperature of 165°F.
Notes
Time for smoking chicken depends on the thickness of the breasts. Use a digital, instant-read thermometer to gauge doneness.