Trim away excess skin and fat from chicken but leave some skin on to brown. Heat 1 tablespoon olive oil (or other cooking oil) in large dutch oven or stovetop-safe slow cooker insert over medium heat on stovetop. Add chicken and lightly brown on both sides. Remove chicken to plate and set aside.
Add a fresh tablespoon of olive oil to pan, add the sliced onion and bell pepper strips, and cook until softened, about 5 minutes. Return the browned chicken and any juices to the pot. Pour in the broth and sprinkle in the cumin, oregano, and salt. Transfer insert to slow cooker, cover, and cook on low heat setting for 3-4 hours until chicken is well cooked and falling off the bone.
Transfer chicken pieces (and the vegetables you can catch) to a cutting board using a slotted spoon, and put the pot of broth on the stovetop, medium-high heat (if your slow cooker insert is not stovetop safe, transfer the broth to a pot); simmer to reduce the broth slightly and concentrate the flavors.
Remove the skin and bone from each piece of chicken, shred or chop the meat, and return it to the broth to serve.
Serving options: roll up the filling in tortillas (or serve as tacos or tostadas) and garnish with shredded cheddar or cotija, avocado or guacamole, cilantro, chopped white onions, pickled jalapeño, sour cream, hot sauce, anything you like that brings Mexican flavors to life.