Trim the pork roast: Remove any large pieces of excess fat from the pork shoulder. Pat dry with paper towels.
Dice and mince: Prepare the onion, red bell pepper, and garlic.
Layer veggies and beans: In a large 6- to 8-quart slow cooker, add the diced onions, bell pepper, minced garlic, diced tomatoes with chilies, black beans, and kidney beans.
Add tomato paste and spices: Pour in the tomato paste, chili powder, cumin, smoked paprika, oregano, salt, black pepper, and cayenne (if using).
Place the pork roast on top: Nestle the pork roast over the layered ingredients, then pour in chicken broth around the sides. Drizzle olive oil over the pork.
Cook: Cover and cook on LOW for 8-9 hours, or on HIGH for 5-6 hours, until the pork is fork-tender and shreds easily.
Shred the pork: Remove the pork roast and place on a cutting board. Shred the meat with two forks, discarding any large pieces of fat, then return shredded pork to the slow cooker.
Finish the chili: Stir in lime juice and adjust seasoning as desired. Let chili cook another 15-20 minutes for flavors to meld.
Serve: Ladle chili into bowls and garnish with chopped fresh cilantro. Enjoy hot, with your favorite toppings.