In a large stockpot over medium heat, add the olive oil and chopped onion, finely chopped carrots, and minced garlic. Cook for about 8 minutes or until the onions are translucent and the carrots are tender.
Add the chicken stock to vegetables and increase heat to medium-high; bring to a boil. Add rice and reduce heat to medium-low; cook, stirring occasionally, until rice is nearly done, about 20 minutes.
In a medium bowl, whisk together lemon juice and eggs. While whisking constantly, slowly pour 1 cup of hot broth into the egg mixture, about a tablespoon at a time. Once all the stock is combined with the egg mixture, pour the egg mixture back into the stockpot.
Add the chicken and salt to the soup, and cook for about 5 minutes. Turn the heat off and add all the baby spinach. Gently stir until the spinach is slightly wilted.
Serve immediately and serve with fresh cracked black pepper if desired.