1. If using raw chicken, place it in a large pot and cover with cold water. Bring to a simmer, skimming off any foam. Lower heat, cover, and cook until chicken is tender (about 45 minutes for thighs, 20 minutes for breasts). Remove chicken, shred meat, and discard bones. Reserve broth and add additional water or broth to reach 6-8 cups if needed.
2. Heat olive oil in a large pot over medium heat. Sauté carrots, celery, and green onions for 3–5 minutes until softened. Add garlic and cook for 1 minute. Pour in chicken broth, add bay leaves, and bring to a rolling boil. Stir in rinsed rice, salt, and pepper. Reduce heat and simmer about 15–20 minutes until rice is tender.
3. Stir shredded chicken back into the soup.
4. In a medium bowl, whisk eggs and lemon juice (and zest if using) until combined. Slowly ladle 1 to 1.5 cups of hot broth into the egg-lemon mixture while whisking continuously to temper the eggs.
5. Remove soup pot from heat and slowly pour tempered egg mixture into the soup, stirring gently to achieve a creamy, silky texture. Avoid boiling after adding eggs.
6. Garnish with fresh parsley or dill and additional lemon wedges if desired. Serve hot with crusty bread or a Greek salad.