1. In a large pot or Dutch oven, heat olive oil over medium-high heat. Sauté the carrots, celery, and green onions (or onion) briefly until slightly softened, then add garlic and cook for another minute.
2. Add chicken broth and bay leaves, bringing the mixture to a rolling boil. If starting with raw chicken thighs or breasts, add them now and simmer on low heat until the chicken is tender and cooked through (about 45 minutes for bone-in thighs, 20 minutes for breasts). Remove chicken, shred meat, and discard bones.
3. Add rice to the broth, along with salt and pepper, and reduce heat to medium-low. Simmer uncovered until rice is tender (about 15 to 20 minutes).
4. In a separate bowl, whisk together eggs and lemon juice. Temper the egg-lemon mixture by gradually whisking in 1-2 ladles of hot broth to prevent curdling.
5. Remove the pot from heat and gradually stir the tempered egg-lemon mixture into the soup while whisking continuously for a smooth, creamy texture.
6. Return the shredded chicken to the pot and stir to combine. Adjust seasoning with additional salt, pepper, or lemon juice as desired.
7. Garnish with fresh parsley or dill and serve hot, ideally with crusty bread or a side Greek salad.