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Wash chicken and remove the inside parts. Rinse with cold water. Add whole chicken to a large pot. Place enough cold water in the pot to cover the chicken and 2 more inches of water. Add 1/2 of an onion, and one carrot cut in thirds, plus one celery stalk.
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Bring to a boil, and reduce to a simmer and cover. Skim off any foam as it appears. Simmer for 2 hours. Remove chicken and place in a bowl.
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Strain broth and return to the pot. If serving with the egg-lemon sauce, remove 1 cup broth and set aside.
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Add 2 teaspoons salt to the broth and bring to a boil.
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Add the orzo, and cook on medium-high for twenty minutes or until orzo is ready.
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Peel and clean the chicken. Cut 1/2 the chicken in small pieces and add to the broth.
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Remove broth from the heat. Make the sauce.
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Gently add the sauce to your broth, and serve.
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Serve with a dash of pepper and a squeeze of fresh lemon.
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Place remaining chicken on a platter and serve with the soup.