Combine the hot water and the salt (brine ingredients) in a 54-quart cooler. Stir until the salt dissolves. Stir in the vegetable broth and the honey. Add the ice and stir. Place the pork in the brine, fat cap up, and cover with cooler lid. Brine overnight, up to 12 hours.
Remove the pork from the brine and pat dry thoroughly. Rub the butt(s) with the vegetable oil. Then add your favorite rub mixture (I included recipe for mine above). Wrap in plastic wrap and allow to sit in fridge at least 3 more hours. Then remove and rest at room temperature for 1 more hour while you pre-heat the grill to 250 degrees F.
Using a smoker, place pork over indirect heat in smoker with hickory wood, (if you have one, insert a probe thermometer into the thickest part of the pork, and set the alarm for 165 degrees F. If you don't have one, you will have to use a stick thermometer every hour after about 10 hours.) Close the lid and cook for at least 4 hours.
After 4 hours check the pork; if the surface is golden brown, cover with aluminum foil tent and continue cooking. Once the butt(s) reach 165 degrees F (about 14 hours), remove from grill, cover loosely with aluminum foil, and allow to rest for 1 hour. Shred then add favorite bbq sauce. (The best brand is Sven’s Sizzlin Sauce, but if you can't find that Famous Dave’s Sweet and Zesty will work as well.).
For my homemade Carolina Style bbq sauce (also good as a marinade):
5 cups cider vinegar 5 tbl kosher salt 2 ½ tsp cayenne pepper 1 tbl + 2 tsp crushed red pepper flakes ¾ cup brown sugar.
Combine and allow to sit at room temperature for at least 6 hours to let flavors blend before using. I usually make it just before I load the meat into the smoker. Then add this sauce to moisten the shredded meat about 1 cup at a time until meat is wet enough to your liking. (About 3½ to 4 cups per 10 pounds of meat for me, you may not like as much or more, here’s a hint: taste often until it’s the way you like it.) Then you can add some of the store brand sauce of your choice, to “kick it up a notch”, if you wish.
Then, I usually finish it by adding a little more of the dry rub. Feel free to experiment with this one, you may or may not want to add any more once the sauce is applied.