Place the chicken, salt, onion, carrots, celery, and water into a stock pot. Bring to a boil over high heat, then reduce heat to medium-low, and simmer until the chicken is tender, about 1 1/2 hours. Skim away any scum that forms as the chicken simmers.
Strain and reserve the broth in a clean pot. Pick the meat from the chicken, discarding the bones, skin, and vegetables. Cut the meat into small pieces, and set aside. Bring the broth to a boil over medium heat and stir in the orzo pasta. Let simmer 20 minutes.
While the orzo is cooking, beat the egg whites to stiff peaks, then whisk in the egg yolks until completely incorporated. Measure out 2 cups of the simmering broth, and combine it with the lemon juice. Slowly pour the lemon mixture into the eggs while whisking continuously until incorporated.
When the orzo is nearly done, return the chicken meat to the soup, and cook until reheated. Stir in the egg mixture, stirring gently until incorporated into the soup. Serve immediately.