In a large pot, combine the whole chicken or boneless chicken, halved onion, smashed garlic cloves, bay leaves, and 10 cups of water or chicken stock. Bring to a vigorous boil over medium-high heat, then reduce to low and let simmer for about 45 minutes.

Once the chicken is cooked, carefully remove it from the pot and set aside to cool. Discard the onion, garlic, and bay leaves back into the broth. Shred the chicken into bite-sized pieces.

With the broth still simmering, add 1 cup of white rice or orzo. Stir occasionally to prevent sticking and cook for about 10-15 minutes until tender.

In a separate bowl, whisk together the 3 large eggs until frothy. Squeeze in the 1/2 cup of fresh lemon juice, blending it into the eggs. Gradually temper the egg-lemon mixture by slowly adding about 1 cup of hot broth.

Lower the heat to a gentle simmer. Whisk the tempered egg-lemon mixture back into the soup, stirring continuously.

Stir in the shredded chicken back into the soup, allowing to warm through. Taste and season with salt and pepper. Serve hot, optionally garnished with fresh parsley or dill.