Simmer whole chicken in water with onion, celery, and carrots 1.5 hours. Remove chicken, strain broth, and shred all meat, discarding skin and bones.
Return broth to pot. Bring to a gentle simmer and add cauliflower rice. Cook 5 minutes.
In a bowl, whisk eggs until frothy. Slowly whisk in lemon juice.
Ladle 1 cup of hot broth into the egg-lemon mixture while whisking constantly — this tempers it so it doesn't scramble.
Pour the tempered mixture back into the pot, stirring gently. Add shredded chicken. Reduce heat to low and cook 2 minutes. Do not boil after adding eggs.
Season with salt and white pepper. Serve garnished with fresh dill.