In a pot over medium heat, combine a quarter of the water with the salt, lemon, thyme, rosemary, brown sugar, garlic and black peppercorns. Simmer until the salt has fully dissolved, about 5 minutes. Remove from heat and cool.
Once completely cooled, combine the brine with the remaining water in a large enough container to hold the water and turkey (that will fit in your refrigerator).
Submerge the turkey in the brine. Refrigerate for 12-20 hours (not longer than this!).
Remove the turkey from the brine and pat dry with a paper towel. Set the turkey on a tray that can be placed in the refrigerator and allow it to dry for another 4-12 hours. (This step can be skipped, but this is how you get the skin really crispy).
Coat the turkey with the olive oil and sprinkle on the dry rub. When ready to smoke, set the smoker temperature to 250˚F, stock with wood pellets or chips (we like to use apple or cherry wood) and preheat for 15 minutes with the lid closed.
Insert the probe into the thickest part of a turkey thigh. Place the roasting pan on the grill grates, close the lid, and roast until the internal temperature reaches 165˚F. It will take about 30 minutes per pound, depends on the size of your turkey. You can place the turkey directly on the grates if you don’t care to catch the drippings in a roasting pan.