Preheat grill over high heat until it reaches 325 degrees then turn half the burners off to create an indirect heat cooking zone, and the other half down to medium-high to maintain the temperature. Combine all rub ingredients in a small bowl then mix to combine.
Pour out half the can of root beer then add 1 teaspoon rub mixture into the can while holding it over the sink as it will foam and over flow a bit. Place root beer can on a plate then set aside.
Pat chicken dry with paper towels then rub all over with oil. Season the inside with a couple big pinches of the rub then season the outside of the chicken (may not need it all.) Place the chicken over the can of root beer, then place inside a disposable pie tin to catch the drippings. Transfer the pie tin to the grill over the indirect heat side - meaning, don't place the chicken directly over the flame.
Put the lid down then grill for 45 minutes — 1 hour and 15 minutes, or until an instant-read thermometer placed into the thickest part of the breast reads 160-165 degrees, keeping the temperature between 325 and 340 degrees during cooking time (it doesn't have to be precise.) Do not lift the lid for at least the first 40 minutes as the grill will take awhile to heat back up every time you open the lid and we want to maintain an oven-like atmosphere inside.
Use tongs to transfer the chicken and root beer can to a plate then let rest for 20 minutes before slicing and serving. Be very careful as the root beer can will be HOT!