I wound up finding a good recipe for Chicken Egg Lemon Soup in a cookbook my sister had. I mad a pot tonight, and it came out really good. I've posted the recipe below. I wound up using orzo instead of rice, and onion powder instead of the finely chopped onion, as my family doesn't like the texture of onions. It's a really good, hearty soup.
Chicken Egg Lemon Soup
6 Cups Chicken broth
1 Chicken flavored bullion cube
1/3 cup regular uncooked rice
1/3 cup diced carrots
1/3 cup chopped celery
¼ cup finely chopped onion
1 cup cubed cooked chicken
2 tablespoons of margarine or butter
2 tablespoons of flour
3 eggs
3 tablespoons of lemon juice
Combine broth, bullion, carrots, celery, onion, and chicken in a large saucepan. Bring to a boil, cover, and simmer for 20 minutes.
In a small saucepan, melt margarine or butter. Stir in flour. Cook for one minute until smooth and bubbly. Gradually stir in two cups of broth mixture. Cook until slightly thickened.
In a small bowl, beat eggs until foamy. Gradually stir in lemon juice and two cups of the thickened mixture. Slowly add everything together. Heat gently until it coats spoon. Do not boil.