In a large pot or Dutch oven, bring the 8 cups of chicken broth to a simmer over medium heat. Add the chicken breasts. Cook for 15-20 minutes, until the chicken is cooked through. Remove the chicken from the pot and set it on a cutting board to cool.

Once the chicken is cool enough to handle, shred it using two forks. While you shred the chicken, bring the broth back to a rolling boil. Add the orzo and cook according to package directions, typically 8-10 minutes, until al dente.

In a medium-sized bowl, add the 3 large eggs. Whisk them vigorously until they are pale yellow and frothy, about 1-2 minutes. Continue to whisk while slowly pouring in the 1/2 cup of fresh lemon juice until fully combined.

Reduce the soup's heat to low. While whisking the egg mixture continuously, slowly ladle about 2 cups of the hot broth from the pot into the bowl. This is the tempering process and it's crucial to do it slowly to avoid scrambling the eggs.

Pour the warm egg-lemon mixture back into the pot with the orzo and remaining broth. Add the shredded chicken. Continue to stir gently over low heat for 2-3 minutes as the soup thickens into a creamy consistency. Do not allow the soup to boil.

Remove from heat. Season with salt and pepper to taste. Stir in the fresh dill. Ladle into bowls and serve immediately.