Cut up chicken into large pieces and place in a large soup pot with the water. Chop 8 carrots, 3 stalks of celery and 1 onion and place them in the pot as well. Bring to a boil then cover, reduce heat and simmer 45 minutes, until chicken is tender. Remove chicken and reserve the stock. When the chicken is cool enough to handle, remove the meat and cut it into bite-size pieces.
Process reserved stock in a blender or food processor or using an immersion blender and return it to the pot with the chicken meat. Chop the remaining 3 carrots, 3 stalks of celery and 1 onion and stir into the chicken mixture with the rice. Bring to a boil, then reduce heat and simmer until rice is tender, about 20 minutes.
Remove chicken mixture from heat and stir in spinach, Romano, salt, pepper, lemon juice and oregano. Set aside.
In large saucepan, bring liquid chicken stock to a boil. Pour the eggs into the boiling stock, slowly, in a thin stream. Remove the stock from the heat and stir it into the chicken mixture. Serve hot.