At least one day before cooking, place fresh turkey on a sheet pan with wire rack and pat dry with paper towels. Cover generously with kosher salt. Refrigerate uncovered for up to 3 days.
Remove from the refrigerator. With a sharp knife, remove excessive neck skin and tailbone. Season generously with black pepper and additional salt if desired. Stuff the cavity with rosemary and orange.
With the turkey breast side up, tuck the wingtips under. Flip the turkey over. Run butcher twine under the neck and over the wingtips. Bring the twine back towards the tail and flip the turkey over so it is now breast-side up. Make a loop in the twine and secure the legs together, tying tightly.
Prepare the grill for indirect rotisserie cooking over medium heat (350º-450ºF) with briquettes pushed to only one side and a foil plan placed in the middle.
Mount the turkey in the center of the rotisserie spit and secure with forks.
Grill the turkey on the rotisserie, keeping the lid closed, until the turkey breast reads 165º F internal with an instant-read thermometer, approximately 2 1/2 to 3 hours. Add unlit briquettes every hour, if needed, to keep the temperature up.
When cooked, remove from the grill, and remove the forks and spit. Carve and serve.