1. Begin by heating olive oil in a large pot over medium heat. Add diced onions and minced garlic, cooking until they become translucent and aromatic. If using celery and carrots, add them now and stir frequently to ensure even cooking.

2. Pour chicken broth into the pot and add your choice of chicken pieces along with a bay leaf. Bring the mixture to a gentle simmer and allow it to cook until the chicken is tender and fully cooked through.

3. Remove the chicken from the pot and let it cool slightly. Shred or chop the chicken into bite-sized pieces and return it to the pot. Add rice and let the mixture cook until the rice becomes tender, about 20 minutes.

4. In a separate bowl, whisk together eggs and freshly squeezed lemon juice until well combined. Gradually temper the egg-lemon mixture by adding a few ladles of the hot broth into it while whisking continuously to prevent scrambling.

5. Slowly pour the tempered egg-lemon mixture back into the soup. Stir gently to incorporate, and heat the soup without boiling to ensure a smooth, velvety texture.

6. Season the soup with salt and black pepper to taste. Serve warm, garnishing with freshly chopped dill or parsley for a fresh and aromatic finish.