Everyone has a different approach, which is great. I've been refining mine for 10 years or so and am really happy with it. My short take is 1) brine, 2) baste, 3)smoke, 4) cook low and slow.
My approach might be seen as labor intensive and overkill, but I like tending the grill, grilling is fun. IMHO, slow cooking is the key to a superior bird. It mimics the cooking in hot rocks method (also a great bird) except that you get to enjoy the added grill & smoke flavors. The short take was above, here is the long version...
What you need:
Turkey, meat thermometer, butter, olive oil, 2 tbsp allspice, 2 cups Kosher salt, 1/4 cup Succanat or brown sugar, rosemary (dry and fresh (if you have it) as specified below (otherwise use dry in all cases), thyme, 1 tbsp peppercorns, sage, charcoal grill with a rack that has liftable flaps on opposite sides, medium bag of charcoal, drip pan – use a tall, disposable roasting pan from the grocery, chips for smoking (hickory or mesquite), food-grade container for brining such as a 5-gallon HDPE bucket with a lid.
2 Days Prior:
Prepare brine – bring 2 gallons water, 2 cups kosher salt, 1/4-cup light brown sugar just to the boiling point. Add 2 tbsp allspice, lot’s o’ fresh rosemary 1 tbsp sage & 1 tbsp thyme and some cracked peppercorns. Cover, remove from heat and let cool overnight.
Don’t even think about stuffing the bird – it works but requires the turkey to be overcooked, regardless of what method you are using to cook it.
1 Day Prior:
In the morning, place brine in refrigerator.
Check to be sure inside of turkey is thawed. It’s helpful to pull out neck and organs.
In evening before bed, rinse bird thoroughly, place in clean brining bucket and cover with chilled brine and cover. If cold outside, leave bucket somewhere safe from animals overnight. Otherwise, store in refrigerator.
After 8-10 hours of brining, pull the bird from the brine and rinse very thoroughly. Then pat dry and allow to air dry until cooking.
Clean the grilling rack.
Soak in water a half of a small bag of hickory chips. Put a sieve on the top to hold them in the water (a pot with a steamer works well).
For planning, cooking time is about 15-20 minutes per pound, but prepare for some flex time.
Prepare a good amount of baste with fresh rosemary & thyme butter by heating on low: unsalted butter, herbs and a little olive oil (two to three sticks of butter and 1/2 to 3/4 cups of olive will make enough). After 20 minutes or so, turn off heat and let this cool until just warm.
Baste between legs and body and tie the legs together. Baste the sides of the breast and inside of wings and tie the wings together.
Insert a thermometer between a thigh and the body (not touching bone).
Here’s a picture of the grill set up (but wait to set it up):
The drip pan will be in the center of the bottom (charcoal) rack. The grilling rack will be placed so that the solid part of the rack sits directly over and aligned with the drip pan. With the proper set up, the flaps can be opened during grilling, allowing you to place coals along the long sides of the drip pan with barbeque tongs. The turkey will be placed so that it is aligned with the drip pan and the flaps can be lifted.
Brush the rack with olive oil and set aside. Start 30 - 50 coals, depending on kettle size. See your Weber cooking instructions supplied with the grill for the number to use. Once they’re turning gray, simply move the charcoal racks to either side. If you are using a chimney to start the coals, empty them from the charcoal chimney onto the bottom (charcoal) grill rack using barbeque tongs. Place 15 coals each in two narrow bands on either side of the rack, directly opposite each other. Work the ash remover to clear out as much potentially airborne ash as possible. Set the aluminum drip pan in the middle of the rack, between the coals. Place the cooking rack on the grill so that the side flaps are positioned directly over the coals. Leave the bottom vents open. Put lid on grill and close the top vents 1/2 to 3/4 (i.e. leave them open about 1/4 to 1/2).
Baste bird and if desired and dust with powdered (dried) rosemary, thyme and sage (do not salt – the meat will be salty from the brine).
Open grill, place the bird on the rack so the rack flaps can be opened and replace lid.
Check the fire every 1/2-hour or so, adding coals through the flaps with barbeque tongs to keep the original amount of fire going. Add coals judiciously; low and slow is the key. You will add up to 8 coals per side each time, depending on how the fire is burning. Baste the turkey after you add the coals. Just before you put the lid back on, you can add presoaked (hickory or mesquite) chips on each side. For a while it will appear that not much is happening, but as long as some coals are alive on each side, it is cooking. The wet chips can be particularly helpful if you think you’ve added too many coals and the fire is getting too hot. This is a preferable method as using a sprayer tends to send ash onto the turkey and into the drip pan. But, cooking too fast makes for a different bird. It’s OK to spend time grilling, this is part of the enjoyment...it is the enjoy the road as well as the destination approach.
If the turkey starts to get too dark, place a piece of foil over it. Be aware that the smoking will make the turkey much darker than usual, but this is different than the darkness caused by heat.
Pull the bird when it reaches 175 degrees (earlier than some say, but 182 is overdone) or just when the legs just start to get loose in the joints (however this is overdone) and let it rest for at least a 1/2-hour.
If the drip pan survives in a clean state, you can use this as a base for gravy.