Heat olive oil in a large soup pot over medium heat. Add diced carrots, onion, and minced garlic. Sauté until tender and fragrant, for exactly 10 minutes. Be careful not to burn the garlic.
Pour in the chicken broth and bring to a simmer. Add the whole wheat orzo and cook for 5 minutes until softened.
Stir in the shredded cooked chicken and remove the pot from heat.
In a small bowl, whisk together the eggs and fresh lemon juice for 30 seconds.
Slowly temper the egg mixture by gradually whisking in one ladleful of hot soup over 1 minute. Then, slowly pour the warmed egg mixture back into the soup pot, stirring constantly until the soup is smooth and creamy.
Stir in the fresh spinach, 1 teaspoon salt, 1/2 teaspoon pepper, and 1 tablespoon dill. Adjust seasonings to taste and serve immediately.