Preheat your smoker to 225°F (107°C). While it heats, combine the salt, garlic powder, onion powder, paprika, brown sugar, thyme, black pepper, and poultry seasoning in a small bowl.
Pat the turkey breast completely dry with paper towels. Rub the olive oil all over the surface of the turkey.
Sprinkle the spice mixture evenly over the turkey, pressing it into the meat. Make sure to cover all sides.
Place the seasoned turkey breast directly on the smoker grate, skin-side up. Smoke for 90 minutes.
After 90 minutes, brush the turkey with the melted butter. Increase the smoker temperature to 350°F (177°C).
Continue to cook until a meat thermometer inserted into the thickest part of the breast reads 165°F (74°C). This will take another 60 to 90 minutes, depending on your smoker and the size of the breast.
Remove the turkey from the smoker and place it on a cutting board. Let it rest for 15 to 20 minutes before slicing.
Slice the turkey against the grain to serve.