Heat oil in a large soup kettle. When oil shimmers and starts to smoke, add chicken breast halves; sauté until brown on both sides, about 5 minutes. Remove and set aside,.
Add half of chopped onions to kettle;sauté until colored and softened slightly, 2-3 minutes. Transfer to medum bowl; set aside.
Add half of chicken pieces; sauté until no longer pink, 4-5 minutes. Transfer to bowl with onions.
Sauté remaining chicken pieces. Return onions and chicken pieces (excluding breasts) to kettle. Reduce heat to low, cover, and simmer until chicken releases its juices, about 20 minutes. Increase heat to high; add boiling water along with both breast halves, 2 teaspoons salt, and bay leaves. Return to a simmer, then cover and barely simmer until chicken breasts are cooked and broth is rich and flavorful, about 20 minutes.
Remove chicken breasts from kettle; set aside. When cool enough to handle, remove skin from breasts, then remove meat from bones and shred into bite-size pieces; discard skin and bone. Strain broth; discard bones. Skim fat from broth, reserving 2 tablespoons (broth can be covered and refrigerated up to 2 days).
Return soup kette to medium-high heat, Add reserved chicken fat. Add remaining onions, along with carrot and celery; sauté until softened, about 5 minutes. Add thyme, along with broth and chicken; simmer until vegetables are ender and flavors meld, 10-15 minutes. Add noodles and cook until just tender, about 5 minutes. Adjust seasonings; sir in parsley, and serve.