Saute. Heat vegetable oil over medium heat in a large stockpot. Once shimmering, add the celery, onions, and carrots, and saute until softened, about 8 minutes. Add the garlic and saute until fragrant, about 2 minutes.
Simmer. Add the broth, chicken, salt, pepper, rosemary, and thyme. Make sure the chicken is fully submerged in broth, and bring soup to a boil. Reduce heat to a simmer, and continue to cook until the chicken reaches an internal temp of 165 -- about 20-25 minutes.
Chicken. Remove cooked chicken to a cutting board and allow it to cool. Once cool enough to handle, shred or chop into bite-sized pieces.
Finish + serve. Stir in the cabbage and cook until softened to your liking (this was 10 minutes for us). Remove soup from heat and stir in the chicken, parmesan, and parsley. Taste and adjust seasoning to your liking. Enjoy!