Place the chicken pieces, onion, carrots, celery, and salt in a large stockpot and add 10 cups of water.
Bring to a boil, then reduce heat to low and simmer for 45-50 minutes until the chicken is tender and fully cooked. Periodically skim any foam from the surface.
Remove the chicken and vegetables from the pot. Strain the liquid through a fine-mesh sieve and return exactly 8 cups of the clear broth to the pot.
Shred the chicken meat, discarding all skin and bones, and set aside.
Bring the strained broth back to a gentle boil, add the rice, and simmer for 15-18 minutes until the rice is tender.
In a medium bowl, whisk the eggs vigorously for 2 minutes until they are frothy and have significantly increased in volume.
Slowly whisk the lemon juice into the beaten eggs until well combined.
To temper the eggs, slowly ladle 2 cups of the hot broth from the pot into the egg-lemon mixture in a thin stream, whisking constantly to prevent curdling.
Remove the stockpot from the heat and slowly pour the tempered egg mixture back into the soup, stirring continuously to create a silky, thickened texture.
Add the shredded chicken back to the pot, season with black pepper, and serve immediately garnished with fresh dill. Note: Do not allow the soup to boil after adding the eggs, as it will separate.