Bring chicken broth, onion, carrot, bay leaf and salt to a simmer. Add chicken breasts, cover and gently simmer over low heat for 30 minutes. Remove chicken breasts, let cool slightly and pull or cut chicken into thin strips. Set aside. Strain soup and return to boil. Add rice and gently simmer for 20 minutes, then add chicken back to soup for another 5 minutes.
Whisk eggs until they are frothy, and whisk in lemon juice. Add a cup of the hot soup to egg-lemon mixture, very slowly, whisking while you pour it in. Add another cup in the same manner. Take soup off the heat and add egg mixture to it. Sprinkle with dill or the parsley, season with salt and pepper and serve