To poach the chicken, place all ingredients into a pot and cover with water. Bring to a boil, reduce heat and simmer for 1 hour, turning occasionally. Let cool in the poaching liquid.
Remove chicken and discard skin and bones. Pull meat using fork tines or your fingers to finely shred.
Strain chicken poaching liquid and reserve about 1 1/2 cups. Refrigerate or freeze remainder for other uses.
While chicken poaches, whisk up slaw dressing in a large bowl. Add cabbage, onion and carrot, toss with dressing and chill.
For barbecue sauce, heat oil, 2 turns of the pan, in deep sauté pan over medium heat. Add onions and garlic, season with salt and pepper, and cook to soften. Add dry spices and toss 1-2 minutes; add liquids and simmer 15 minutes to thicken a bit. Add in pulled chicken and toss to coat.
Serve chicken piled on soft rolls topped with slaw and pickles.