Heat oil in a soup kettle over medium-high heat. Add chicken breast halves and sear until browned on each side, about 5 minutes. Remove and set aside.
Add half the onions to the kettle, sauté until softened, 2–3 minutes. Remove and set aside.
In batches, sauté half the remaining chicken pieces until no longer pink, 4–5 minutes. Repeat with the rest of the chicken.
Return all sautéed chicken and onions to the kettle. Cover, lower heat, and cook for 20 minutes.
Add boiling water, chicken breast halves, salt, and bay leaves. Bring to a simmer, cover, and cook for 20 minutes.
Remove chicken breasts, cool, discard skin and bones, shred meat. Strain broth, discard solids, and skim fat, reserving 2 tablespoons.
Return kettle to medium-high heat, add reserved fat, remaining onions, carrots, and celery. Sauté for 5 minutes.
Add thyme, strained broth, and shredded chicken back to the kettle. Simmer for 10–15 minutes.
Stir in noodles, cook until tender, about 5 minutes. Adjust seasoning, stir in parsley, and serve hot.