If using raw chicken, place chicken thighs in a large pot with chicken broth. Bring to a boil, then reduce heat and simmer for 25-30 minutes until cooked through. Remove chicken, shred the meat, and discard bones. Reserve the broth.
In the same large pot or Dutch oven over medium heat, sauté the diced onion for 5 minutes until softened. Add the minced garlic and cook another minute until fragrant.
Add the crushed tomatoes, diced tomatoes, reserved chicken broth (or 2 cups fresh broth), BBQ sauce, Worcestershire sauce, brown sugar, smoked paprika, salt, pepper, and cayenne. Stir to combine.
Add the shredded chicken, pulled pork, corn, and lima beans. Stir everything together and bring to a simmer.
Reduce heat to low and simmer uncovered for 45-60 minutes, stirring occasionally, until the stew has thickened and the flavors have melded. Taste and adjust seasoning. Serve hot with cornbread.