Prep the sauce base: In a bowl, whisk together BBQ sauce, tomato paste, apple cider vinegar, brown sugar or honey, Worcestershire, smoked paprika, chili powder, and chicken broth.
Taste and adjust sweetness, tang, and heat as you like.
Sauté aromatics: In a large pot or deep skillet, warm a bit of oil over medium heat. Add the diced onion and a pinch of salt. Cook 3 to 4 minutes until softened.
Stir in the garlic and cook 30 seconds until fragrant.
Add chicken and sauce: Nestle the chicken into the pot. Pour the sauce over the top and turn the chicken to coat. Bring to a gentle simmer.
Simmer until tender: Cover and cook over low heat for 15 to 20 minutes for breasts, 20 to 25 minutes for thighs, until the chicken is cooked through and easily shreds.
Stir occasionally so nothing sticks. If it looks dry, splash in a bit more broth.
Shred the chicken: Transfer the chicken to a cutting board and shred with two forks. Return it to the pot and toss to coat in the sauce.
Simmer uncovered 3 to 5 minutes to thicken. Season with salt and pepper.
Toast the buns: Spread a little butter on the cut sides of the buns. Toast in a skillet over medium heat until golden.
This keeps them from getting soggy and adds flavor.
Mix the slaw: If making a quick slaw, toss shredded cabbage and carrots with mayo, a splash of vinegar, a pinch of sugar, salt, and pepper. Keep it crisp.
Assemble: Pile the saucy pulled chicken onto the toasted buns. Top with slaw and pickles.
Press gently and serve warm.