Multiply the brine recipe as needed to make enough to completely submerge the turkey breast. Stir until salt and sugar are dissolved.
Put the turkey breast in a large Ziploc bag, fill with brine, squeeze out all the air and seal; or put turkey breast in a large bowl, put a small plate on top to hold it under, and fill bowl with brine. Refrigerate for 24 to 48 hours (no longer, or the meat will be too salty).
Remove turkey from brine, rinse, and dry. Rub lightly with the wet rub. Combine the dry rub ingredients, apply to the bird, cover with plastic wrap and let sit 30 minutes. Slow-cook in a smoke-cooker, or covered barbecue using the indirect heat method, and adding water-soaked smoke chips from time to time. Place a meat thermometer in the thickest part of the breast and cook until it reaches 165 to 170 degrees. If skin looks dry at any point, brush with a little more oil.
For best flavor and appearance, cool to room temperature, slice, and serve at once. Turkey can also be wrapped tightly and refrigerated; return to room temperature before serving.
This comes out a gorgeous mahogany color in a smoker, so it looks great on a buffet table. Brining makes the meat turn out moist and firm-textured. I made it in a regular Weber barbecue before we got the smoker, and it came out a little less smoky-flavored, but still really good. Last year I made four of these for our holiday party, and they disappeared as fast as I could slice them!
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