Add onion to the pot and sauté over medium-high heat until lightly colored and soft, stirring constantly, 2 to 3 minutes.
Find those two breast halves in the bowl of chicken, and set them aside. Return the rest of the chicken to the pot including any juice that has accumulated in the bowl.
Reduce heat to low, cover, and cook until chicken releases its juices, about 20 minutes.
In the meantime, set a kettle or pot of water to boil.
Increase heat under the stockpot to high, add boiling water, the two chicken breasts you've set aside; salt, pepper, and bay leaves. Bring back to barely simmer, cover until breasts are cooked, about 20 minutes. By now the dark broth will be extra rich and flavorful.
Remove the breasts from the pot and set aside. When cool enough to handle, remove the skin and shred the breast meat from the bones. Discard breast skin and bones.
Strain broth into a separate pot and discard the solids. All of the goodness has been cooked from the rest of the chicken (dark meat, skin and bones), so you are not being wasteful. Skim fat for later use in other recipes or discard. Return the clear broth to the pot with the shredded chicken. Bring back to boil.
In a small skillet, sauté the chopped carrot and chopped celery in 1 tablespoon of oil. Add to the pot along with the egg noodles and cook until just tender, about 5 minutes. Adjust with salt and pepper as necessary and serve. Serves 6 to 8.
To your health!