Combine all ingredients in a large pot. Stir until salt and sugar dissolve. Bring to a boil, then turn off heat and cover.
Allow to cool completely, then pour into a large brining bag or pot. Place uncooked turkey in brine solution, then refrigerate for 16 to 24 hours.
When ready to roast turkey, remove turkey from brine. Submerge turkey in a pot or sink of fresh, cold water. Allow to sit in clean water for 15 minutes to remove excess salt from the outside.
Discard brine. Remove turkey from clean water, pat dry, and cook according to your normal roasting method.
Pat turkey breast dry and apply BBQ rub
Set up the Big Green Egg for indirect cooking measured to 250°
Use a drip pan full of water sitting on top of the convEGGerator
Add chunks of a light flavored wood (e.g. Pecan, Apple) for the smoke
Once the temp stabilizes and you see the thin blue smoke place the turkey breast inside
Smoke for 1.5 hours or until the turkey measures 145° internally
Increase the heat in the Big Green Egg to 300° and cook turkey for another 30 minutes or until it measures 155° internally
Remove the turkey breast and let rest for 10 minutes before slicing and serving