Season the chicken with salt and pepper.
Heat a pot over medium heat. Once hot, add the oil. Saute the onion, celery, carrot, garlic, and oregano until the celery starts to soften. Season with salt and pepper.
Stir in the broth, chicken, lemon juice, and rice. Cover with a lid and bring to a simmer until the rice is cooked, about 20 minutes. Refer to the instructions on the package for cooking time.
Once the rice is tender, simultaneously stir a ladle of the broth to into the beaten egg until fully incorporated. Repeat this 1 or two more times to temper the egg so it does not curdle when added to the soup.
Stir the egg into the simmering soup, it will thicken up to a creamy consistency. (add broth if the soup becomes too thick). Add more lemon if needed so there is a lemony flavor but not overpowering. Adjust the seasoning with salt and pepper to taste. Remove from the heat and stir in the parsley.
Garnish with lemon slices and parsley. Serve with crackers.