To serve: thin lemon slices, fresh dill, freshly ground black pepper
Place the chicken, water, onion, celery, peppercorns, and salt in a large stockpot. Bring to a simmer, then lower the heat, cover, and cook at a bare simmer until chicken is cooked through. Remove chicken from stock to cool, strain stock and return to soup pot.
When the chicken is cool enough to handle, shred the meat and set aside.
Bring the broth back up to a low boil, add the orzo and cook until al dente. Stir in the shredded chicken and turn the heat to low.
Crack the eggs in a medium bowl and whisk until frothy. While still whisking, add the lemon in a thin stream. Continue whisking while slowly drizzling in about ½ c hot stock to prevent eggs from curdling.
Add beaten eggs to soup pot and cook over low heat for 3-5 minutes until soup thickens. Take care not to let it come to a boil.
Ladle into soup bowls and serve immediately, garnished with lemon slices, dill, and freshly ground black pepper.