In a large pot, heat oil over medium-high heat. Add onion, carrots, garlic, celery, salt and pepper and cook for 3-4 minutes, until onion is translucent.
Add broth, chicken, and rice. Stir and bring to a boil over medium-high heat.
Once at a boil, reduce heat to medium-low and cover. Simmer for 30 minutes or until vegetables and rice are tender and chicken has reached 165℉.
Remove 1-1½ cups broth into a medium size bowl or large measuring cup and allow to cool a couple minutes - you still want it hot for the egg lemon sauce.
In the meantime remove pot from heat, then remove chicken from pot and chop or shred and return shredded chicken to pot.
Separate eggs and beat egg whites whisking until they are nice and frothy then whisk in the yolks well, followed by the lemon juice and then while whisking gradually pour in the warm/hot reserved broth.
In a slow stream drizzle the avgolemono sauce to the soup pot and gently stir.
Return pot to heat over low heat and continue to cook on low or medium low until soup is fully heated through with a creamy texture. Taste and adjust for salt and pepper.
Ladle out into bowls and serve with a drizzle of olive oil, more fresh lemon juice/lemon wedges and chopped fresh herbs - I recommend fresh dill.
