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Heat the oil in a large heavy-bottomed pot and sauté the breasts until they are light brown, about 5 minutes
Remove the breasts and set aside.
Add the onions and sauté until translucent, approximately 4 to 5 minutes. Remove and set aside
Cut up the remaining parts of the chicken (not the breasts) into small pieces. Add the pieces to the pot, and cook for 8 to 10 minutes until no longer pink
Return the onion to the pot . Reduce the heat to low, cover, and simmer until the chicken releases its juices, about 20 minutes
Add boiling water, reserved chicken breasts, 2 teaspoons of salt, and bay leaves . Cover and simmer until chicken breasts are cooked, about 20 minutes. Increase the heat if necessary
Remove chicken breasts and set aside
Strain and skim the fat from the broth, reserve broth. Reserve 2 Tablespoons of the fat for cooking the vegetables
Add the reserved fat to the soup pot and sauté the remaining onions along with the carrot and celery for about 5 minutes
When the chicken breasts are cool enough to handle, remove and discard the skin and bones. Shred the breast meat into bite sized pieces and add to the pot
Add thyme and reserved broth; simmer until the vegetables are tender
Season with salt and pepper, add parsley and serve