Add the beer (or buttermilk) to a large bowl or ziptop bag. Add in the kosher salt and stir to combine. Add the chicken, submerging all the parts in the beer mixture. Set aside to brine for at least 20 minutes or up to 2 days. If brining longer than 1 hour, be sure to store the chicken in the brine in an airtight container or ziptop bag in the refrigerator. If I have the time, I usually try to brine for at least 12 hours.
Prepare your grill for indirect cooking, creating a zone of direct high heat & a second zone of indirect heat. If using a gas grill, begin preheating at least 15 minutes before grilling. If using charcoal, light the charcoal at least 30 minutes prior to grilling.
Once the grill is hot, remove the chicken from the brine, patting off any excess moisture with a paper towel. Season generously with salt & the BBQ chicken dry rub. If using skin-on chicken, work your fingers under the skin to season both sides of the skin.
Place the seasoned chicken on the indirect side of the grill (skin side up, if using skin-on chicken), as close to the fire as possible without being over the direct heat. Cook the chicken, rotating the pieces 180 degrees every 15 minutes, until an instant-read thermometer inserted in the thickest part of the chicken (without hitting the bone) registers a temperature of 150 degrees F. Depending on the size of your chicken & the temperature of your grill, this could take as little as 15 minutes or up to 40 minutes.
Move the chicken over direct heat. If using skin-on pieces, flip the chicken so the skin-side is down. Use a pastry brush to baste the BBQ chicken with BBQ sauce & cook for 2 minutes. Flip the chicken & cook for 2 minutes more. If using skinless chicken, baste the second side with BBQ sauce; if using skin-on chicken, I suggest not applying BBQ sauce to the skin to keep it nice & crispy. Repeat 2-3 times, until your chicken is as charred & saucy as desired and an instant-read thermometer inserted in the thickest part of the chicken (without hitting the bone) registers a temperature of 160 degrees F. Remove from the grill and set aside to rest for 5-10 minutes.