In a large pot, heat 1 tablespoon of olive oil over medium heat. Once the oil shimmers, add the chicken breasts to the pot. Cook for approximately 5 minutes on each side, or until golden brown. Once cooked, remove the chicken from the pot and set aside on a cutting board to cool slightly.
In the same pot, add the finely chopped onion, diced carrots, and diced celery. Sauté the vegetables for about 5 minutes, stirring occasionally, until they start to soften. Add the minced garlic and continue to sauté for an additional minute, or until the garlic becomes fragrant.
Pour the chicken broth into the pot and bring it to a gentle boil. Once boiling, lower the heat to medium low, reintroduce the cooked chicken to the pot, and allow it to simmer for 15-20 minutes, or until the chicken is fully cooked through.
Carefully remove the chicken once more and place it on a cutting board. Stir the orzo pasta directly into the boiling broth and cook according to the package instructions, which typically takes around 8-10 minutes.
While the orzo is cooking, in a large mixing bowl, whisk together the eggs, freshly squeezed lemon juice, and lemon zest until the mixture is smooth and fully combined.
Gradually ladle about 1 cup of the hot broth from the pot into the egg mixture while whisking continuously. This step helps to temper the eggs, preventing them from scrambling when added to the hot soup.
Slowly pour the tempered egg mixture back into the pot while stirring gently. Shred the cooled chicken into bite-sized pieces and return it to the soup pot. Season the soup with salt and freshly cracked pepper to taste, adjusting as necessary.
Allow the soup to simmer for an additional 2-3 minutes, letting it warm through and slightly thicken. Remove the pot from heat.