In a large pot, heat oil over medium high. Add onion then saute until soft and fragrant, 2-3 minutes.
Add carrots and celery, cooking an additional 3-5 minutes. Add garlic and continue cooking, stirring frequently, another 1-2 minutes.
Add broth, undrained tomatoes, orzo, thyme, oregano, and rosemary. Stir to combine well then bring to a boil.
Reduce heat, cover, and simmer until orzo is al dente consistency.
Stir in the spinach and allow to wilt.
Adjust seasoning to taste and serve warm.