Mix the smoked paprika, garlic powder, onion powder, cayenne pepper, dried thyme, 1 tablespoon salt and 1 1/2 teaspoons black pepper in a bowl. Sprinkle some of the spice rub inside the cavity of the turkey. Separate the skin from the breast meat with your fingers, starting at the top of the breast and sliding to the right and left, then working down. Massage some of the rub onto the meat under the skin. Sprinkle the remaining rub on the turkey's skin. Place the turkey on a sheet tray and cover with plastic wrap. Refrigerate overnight or up to 24 hours so the flavors can marry.
Set a rack at the lowest position in the oven and preheat to 325 degrees F. Remove the turkey from the refrigerator to bring to room temperature. Tie the legs together and tuck the wing tips under. Place the turkey in a roasting pan. Drizzle the outside of the turkey with a few tablespoons of olive oil and sprinkle with salt and pepper. Roast the turkey about 3 hours, or until a thermometer inserted into the thickest part of the thigh registers 165 degrees F. Transfer the turkey to a platter, cover loosely with foil and let rest 30 minutes before carving.