Heat the olive oil in a large Dutch oven over medium heat. Add the onion, carrots, and celery and cook for 7–8 minutes until softened.
Stir in the garlic and cook for 30 seconds. Add the Greek Seasoning, salt and bay leaves. Cook for 1 minute until fragrant. Pour in the chicken broth and add the chicken breasts. Bring to a gentle boil, then reduce to a simmer. Cover and cook for about 15 minutes, or until the chicken reaches 165°F.
Remove the chicken with a slotted spoon and transfer it to a cutting board. Add the rice to the simmering broth and cook for 15–18 minutes, or until tender.
While the rice cooks, shred the chicken using two forks. Return the shredded chicken to the pot and reduce the heat to low.
In a medium mixing bowl, whisk together the whole eggs and egg yolks for about 1 minute until light and frothy. Whisk in the fresh lemon juice.
Slowly ladle about 1½ cups of the hot broth into the egg mixture a little at a time, whisking constantly to temper the eggs.
Very slowly pour the tempered egg mixture back into the soup while stirring constantly.
Continue cooking over low heat for 2–3 minutes until the broth becomes silky and slightly thickened. Do not allow the soup to boil.
Remove the bay leaves. Stir in the fresh dill and parsley.
Taste and adjust with additional salt, black pepper, or lemon juice if desired. Serve immediately with lemon wedges and fresh cracked black pepper.
