In a large pot, heat the olive oil over medium heat. Add the finely chopped onion and sauté for about 5 minutes, or until the onion becomes translucent and soft.
Stir in the minced garlic and continue to sauté for an additional minute, until it releases its fragrant aroma.
Pour in the chicken broth and bring the mixture to a gentle boil, allowing the flavors to meld together.
Carefully add the chicken breasts to the pot. Reduce the heat and let it simmer for 15-20 minutes, or until the chicken is thoroughly cooked. After cooking, remove the chicken from the pot and set it aside to cool slightly.
Once the chicken is cool enough to handle, shred it into bite-sized pieces and return it to the pot, allowing it to regain its warmth.
Stir in the orzo pasta and let it cook in the bubbling broth according to the package instructions (usually about 8-10 minutes), or until it's al dente, stirring occasionally.
In a separate bowl, whisk together the eggs, fresh lemon juice, and lemon zest until you achieve a smooth consistency.
To prevent the eggs from scrambling, gradually introduce a ladle of hot broth into the egg mixture while whisking continuously to raise its temperature.
Slowly pour the tempered egg mixture back into the pot, stirring gently as you add it to ensure a creamy texture throughout the soup.
Taste the soup and adjust the seasoning with salt and pepper as desired. Stir until everything is well combined and heated through.
Serve the soup hot, garnished with a sprinkle of fresh dill or parsley for a burst of color and flavor.