Remove the neck and giblets from the turkey. Rinse inside of turkey; pat dry with paper towels. If present, remove and discard the plastic leg holder and pop-up timer.
Preheat oil to 350 degrees F.
For rub, in a small bowl combine salt, paprika, thyme, black pepper, garlic powder, onion powder, and cayenne pepper. Slip your fingers between turkey skin and meat to loosen the skin over the breast and leg areas. Lift skin and spread some of the rub directly over breast, thigh, and drumstick meat. Season body cavity with any remaining rub. Tuck the ends of the drumsticks under the band of skin across the tail or tie legs to tail with 100-percent-cotton string. Twist wing tips under back.
Place turkey, breast side up, in basket. Slowly lower basket into hot oil. Be cautious of splattering oil. Maintain oil temperature around 350 degrees F. Fry turkey for 24 to 30 minutes (3 minutes per pound). Remove turkey from hot oil to check doneness. Insert a meat thermometer into the meaty part of the thigh. Turkey is done when thermometer registers 180 degrees F.
Remove turkey from hot oil; drain on wire rack. Let stand 15 minutes before carving.