In a pot with water, bring chicken, onions, carrots, celery, bay leaves and oregano to a boil and then simmer uncovered for 1 - 1.5 hours until bones are easy to separate and chicken is cooked through; remove chicken and discard vegetables

In a skillet, cook onion and salt in olive oil over MEDIUM heat until soft and golden

Turn pot to MEDIUM LOW and add sauteed onions, rice and salt; simmer, stirring occasionally, for 30 - 45 minutes until soft and tender

Cut chicken into bite size pieces

In a bowl, whisk together eggs, pepper, cayenne and lemon juice

Add 1 cup hot broth to the bowl while whisking

Add another cup hot broth to the bowl while whisking

Whisk egg mixture into the soup

Add chicken meat and cook over LOW for 5 minutes

Test for seasoning