Warm olive oil in a large pot over medium heat.
Sauté onion, carrot, and celery until softened, about 5–6 minutes. Stir in garlic for 30 seconds.
Pour in chicken broth and bring to a boil. Add rinsed rice, then lower heat and simmer until tender, 14–16 minutes.
Stir in shredded chicken and warm through on low heat, 3–4 minutes. Do not boil hard.
Whisk beaten eggs, lemon juice, and zest in a medium bowl until smooth.
Temper the egg mixture: slowly whisk in 2 cups hot soup liquid, a ladle at a time, until steamy and creamy.
Take the pot off heat. Stir the tempered egg-lemon mixture back into the soup until silky. Do not let it boil.
Season with salt, pepper, and dill. Let rest 2 minutes, then ladle into bowls and serve.