Make the brine by adding 2 cups of boiling water in a large container with the salt, sugar and peppercorns. Stir to dissolve completely, then add a combination of 6 cups of cold water and ice.
Place the turkey breast in the brine and refrigerate for 8 hours.
Remove the turkey from the brine, rinse and pat dry with paper towels.
Make the glaze
While the turkey smokes, make the glaze. In a small saucepan over medium-low heat, add the glaze ingredients and stir to combine and dissolve. Bring to a simmer and reduce the heat to low, continuing to cook until the glaze has thickened enough to coat the back of a spoon.
Smoke the turkey
While the breast comes to room temperature, prepare the smoker according to the manufacturer’s instructions for indirect cooking with a water drip pan in place. Add wood chunks (or pellets) and bring the temperature to 250° F.
Rub room-temperature butter or canola oil all over the turkey, then place in the smoker.
When the internal temperature of the breasts reaches about 130° F (about 90 minutes in), glaze the turkey and repeat after 15 more minutes.
Remove the turkey at 160° F, as it will continue to rise at least 5° to finish at 165° F. Verify with a probe thermometer.
Rest for 20 minutes before slicing and serving with a drizzle more of the glaze.