Whisk the eggs in a bowl, then run them through a fine-mesh sieve to remove any stray shells.
Combine the 12 cups chicken broth and the lemon juice in a pot and bring to a boil over high heat. Lower the heat and hold at a simmer.
Temper the eggs by gradually whisking broth into the eggs, one ladleful at a time. When you have added about a quarter of the broth, stir the mixture back into the pot. Cook at a low simmer, stirring constantly, until slightly thickened, about 2 minutes. (Be careful not to let the soup boil or the eggs will overcook and ruin it.) Stir in the salt and white pepper and strain the soup through a fine-mesh sieve to remove any lumps.
In a small saucepan, combine the shredded chicken and the 1/2 cup chicken broth. Bring to a simmer to warm the chicken, then strain off the broth. Divide the chicken and orzo among six bowls; add 1 teaspoon of the chopped dill to each bowl. Ladle the soup into the bowls and serve immediately.