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For the Chicken and Rub: Mix together salt, paprika, garlic powder, onion powder, and cayenne pepper in a small bowl. Season chicken halves all over with rub and refrigerate for at least 6 and up to 12 hours.
For the Sauce: Heat oil in a medium saucepan over medium-high heat until shimmering. Add onion and cook, stirring occasionally, until softened and browned around the edges, about 7 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Stir in ketchup, apple juice, brown sugar, honey, vinegar, molasses, Worcestershire sauce, mustard, soy sauce, and hot sauce. Bring to a boil, then reduce heat and simmer until slightly thickened, 15-20 minutes. Remove from heat and let cool slightly.
Puree sauce with an immersion blender, or transfer sauce to the jar of a standard blender, and process until smooth. Season with salt and pepper. If not using right away, transfer to a jar and store in refrigerator.
Fire up smoker or grill to 225°F. Add wood chunk to smoker following manufacturer's instructions, or set wood chunk on top of embers if using a grill. Place chickens in smoker or on grill over indirect heat, and cook until breast meat registers 160°F on an instant read thermometer, 1 1/2 to 2 hours. Remove from smoker and let chickens rest until cool enough to handle.
Using hands or two forks, pull chicken into thin strands, discarding skin, cartilage, and excess fat.
Measure out 3/4 cup of barbecue sauce and set aside. Reheat remaining barbecue sauce in a large saupan. Stir shredded chicken into hot barbecue sauce and cook until chicken is warmed throughout. Serve immediately on sandwich rolls with reserved barbecue sauce and coleslaw, if desired.