Heat the olive oil in a large pot over medium heat. Cook and stir carrots, celery, and onion in hot oil until the vegetables begin to soften and the onion becomes translucent, 5 to 7 minutes. Add garlic; cook and stir until fragrant, about 1 minute more.
Season mixture with thyme, oregano, salt, black pepper, and bay leaf; continue cooking another 30 seconds before pouring chicken broth into the pot.
Bring the broth to a boil. Partially cover the pot, reduce heat to medium-low, and simmer until the vegetables are just tender, about 10 minutes.
Stir in the orzo, then cook until softened. Add the, lemon juice into the broth Cook until the chicken and orzo are heated through, about 5 minutes.
Add the spinach and the chicken; cook until the spinach wilts into the broth and the chicken is heated through, around 4-5 minutes.
Ladle the soup into bowls; garnish with lemon slices and sprinkle with Parmesan cheese.
Use a store bought rotisserie chicken to save time on this recipe!