Warm a large soup pot over medium heat and drizzle in the olive oil. Toss in the diced carrots, onion, and garlic and cook gently until the veggies are soft and fragrant, about 10 minutes—just watch the garlic so it doesn’t brown too much.
Pour in the chicken broth and bring everything to a simmer. Stir in the orzo and keep cooking until it becomes tender, then mix in the shredded chicken and take the pot off the heat.
In a small bowl, whisk together the eggs and lemon juice. To prevent them from scrambling, gradually ladle some warm soup into the egg mixture while stirring.
Slowly pour the egg mixture back into the soup pot bit by bit, stirring continuously to create a silky, creamy texture.
Stir in fresh spinach, salt, freshly ground pepper, and plenty of dill. Taste the soup and tweak the seasoning if necessary before serving.
