Place the sliced onion into the slow cooker. Put the chicken breasts over the onions. Season the chicken with the cumin, chili powder, dried oregano, salt, and pepper. Dot the chicken with garlic.
Pour the green enchilada sauce and green chilies over the chicken in the slow cooker, ensuring the chicken is well coated. Cover and cook on low for 6 hours or until the chicken is tender and cooked through.
Once cooked, shred the chicken using two forks directly in the slow cooker. Serve the shredded chicken with cooked rice or tortillas. Garnish with chopped fresh cilantro and lime slices if desired.