Place the chicken or pork in a slow cooker on low heat. Add all slow cooker remaining ingredients and cook for 8-10 hours. Remove the chicken, discard the bones and skin, and shred the meat using two forks. Place the shredded chicken in a large bowl. Use a slotted spoon to lift out the vegetables in the slow cooker and combine them with the chicken. Discard the liquid.
To make the barbecue sauce: Melt the butter over medium heat in a medium sauce pan. Saute the onions and garlic until soft and translucent, about 7 minutes. Add all remaining ingredients and stir to combine. Bring to a boil, reduce the heat to low, and simmer uncovered for 2 hours, stirring occasionally to prevent burning. Add salt and pepper to taste.
Combine half of the sauce with the shredded chicken or pork and reserve the rest for passing around the table.Assemble the bbq pulled chicken sandwiches, placing sliced pickles on top if desired and some of the extra bbq sauce. Serve with your favorite sides.
*Suggestion: Make the sauce in advance, up to 2 days. It saves time plus it will taste even better.