Halve the chicken by removing the backbone and splitting the breastbone.
Evenly sprinkly the dry rub over the entire chicken and refrigerate overnight, or for at least 2 hours.
Remove the chicken from the refrigerator to bring up to room temperature while you set up the grill for two-zone cooking with the direct side on medium-high.
Lightly oil the grill grates, then place the chicken skin-side down over direct heat with the lid closed until browned. Flip once, cooking about 5 to 7 minutes per side.
Move the chicken halves over to the indirect side of the grill with the breasts farthest from the direct heat and maintain a 400° F temperature with the lid closed.
After cooking for 20 minutes, use a heatproof basting brush to apply about ½ cup of BBQ sauce to both sides of the chicken (finishing with the skin side up).
After 10 more minutes, brush on ¼ cup more BBQ sauce.
Continue to cook until the internal temperature of the breasts reaches 160° F and thighs reach 175° F, about 20 minutes longer. There will be some carry-over cooking, which will raise the finished temperature another 5° or so. (Verify with a probe thermometer.)
Transfer the chicken to a cutting board to rest uncovered for 10 minutes before slicing and serving with more BBQ sauce.
There will be about 5° of carry-over cooking after the chicken is removed from the grill, so I like to pull the chicken when the breasts are 160°.