In a large pot or Dutch oven, combine the chicken broth, whole onion, and whole carrots. Bring to a simmer over medium-high heat.

Carefully add the chicken breasts to the pot. Reduce heat to maintain a gentle simmer and cook for 15-20 minutes, or until the chicken is cooked through. Remove the chicken, onion, and carrots. Discard the vegetables. Let the chicken cool slightly, then shred it using two forks.

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

While the orzo cooks, prepare the Avgolemono sauce. In a medium bowl, whisk the eggs until pale and frothy, about 2 minutes. Slowly whisk in the fresh lemon juice until well combined.

Temper the eggs: While continuously whisking the egg-lemon mixture, slowly pour in about 2 cups of the hot broth from the pot, one ladleful at a time. This will gently raise the temperature of the eggs without scrambling them.

Once the orzo is cooked, add the shredded chicken back to the pot. Reduce the heat to low.

Slowly pour the tempered Avgolemono sauce into the soup pot, stirring constantly. Continue to stir gently over low heat for 2-3 minutes as the soup thickens. Do not let the soup boil.

Season with salt and pepper to taste. Serve immediately, garnished with fresh dill or parsley.