Bring first 4 ingredients to a boil in a Dutch oven; cover, reduce heat, and simmer 1 hour or until tender. Remove chicken, reserving broth in Dutch oven.
Skin, bone, and chop chicken. Return chicken to broth. Add onion and next 6 ingredients.
Bring chicken mixture to a boil; reduce heat, and simmer, stirring occasionally, 1 hour and 30 minutes. Add cheese, and cook, stirring often, until cheese melts (do not boil).