1. Combine all the brine ingredients in a pot large enough to hold the chicken and bring to a simmer, stirring to dissolve the salt and sugar. Remove from the heat and allow to cool to room temperature, then refrigerate until chilled.
2. Remove any giblets from the cavity and truss the chicken, then submerge it in the brine. Weight it down with a plate to keep it submerged, and refrigerate for 18 hours.
3. Remove the chicken from the brine (discard the brine), rinse under cool water, and pat dry. Refrigerate, uncovered, for a least 3 to 4 hours, and up to a day.
4. Combine the glaze ingredients in a small heavy bottomed pan and bring to a simmer over high heat, stirring to dissolve the sugar. Lower the heat and gently simmer until thick and syrupy and reduced to about a cup/250 milliliters/ Remove from the heat and let cool.
5. Begin hot-smoking the chicken. Halfway through the smoking—about 1 ½ hours if you’re smoking at about 200 degrees F./93 degrees C.—remove the chicken from the smoker and brush all over with half the glaze. Return the chicken to the smoker and continue cooking until a thermometer inserted into the thigh registers 160 degrees F./71 degrees C.
6. Remove the chicken from the smoker and brush all over with the remaining glaze. It should have a deep brown color.
7. Serve hot, or cool and refrigerate, covered until ready to serve.