Heat olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion, carrots, and celery and cook until softened, about 5-7 minutes.

Pour in the chicken broth and bring to a simmer. Add the chicken breasts to the pot. Cook for 15-20 minutes, or until the chicken is cooked through. Remove the chicken from the pot and set aside to cool slightly before shredding with two forks.

Bring the broth back to a simmer. Add the orzo and cook according to package instructions, typically 8-10 minutes, until al dente.

While the orzo cooks, prepare the Avgolemono sauce. In a medium bowl, whisk the eggs vigorously until pale and frothy. While still whisking, slowly pour in the lemon juice until combined.

Very slowly, ladle about 1 cup of the hot broth from the pot into the egg mixture while whisking constantly. This tempering process prevents the eggs from scrambling. Repeat with two more ladles of hot broth until the egg mixture is warmed.

Reduce the heat of the soup to low. Slowly pour the warm egg-lemon mixture back into the pot, stirring constantly. Add the shredded chicken back to the soup.

Continue to stir gently over low heat for 2-3 minutes until the soup has thickened. Do not allow the soup to boil. Remove from heat, stir in the fresh dill or parsley, and season with salt and pepper to taste. Serve immediately.