In 2-quart saucepan, bring broth to a boil. Reduce heat, add rice, cover and simmer until rice is tender.
In small bowl, beat eggs and lemon juice. Stirring constantly, add 1 cup of hot chicken broth. Stir mixture into soup in pan and heat. Do not boil! Season with salt and pepper.
Lower heat to simmer; cook, covered, for 20 minutes, stirring often. Remove from heat; stir in butter, slightly beaten eggs, milk, diced cheese, garlic, pepper, and hot sauce.
Place in greased casserole and sprinkle with grated cheese.