Warm the oil in the stovetop-safe insert to your slow cooker, add the chicken breasts, skin side down and brown. Add the onions and cook for a few more minutes. (If your slow cooker has a ceramic insert that is not safe for the stovetop, use a regular frying pan for all this browning stuff (step 1 and 2), then transfer all to your slow cooker and add the broth.)
Add the green chiles, broth and salt. Transfer insert to slow cooker and set timer for 4 hours on High setting.
When chicken is done cooking, transfer the pieces to a cutting board, remove and discard bones and skin. Shred chicken with a fork. (I like to transfer my slow cooker insert with the sauce from cooking to the stovetop over medium-high heat to boil and reduce the sauce while I shred the chicken.) Return shredded chicken to the sauce and serve.