Heat oil in a large stockpot over medium-high heat. Add onion and saute for 4 minutes, until soft. Add carrots, celery and garlic and saute for an additional 3 minutes. Add chicken stock, tomatoes, orzo, thyme, oregano, rosemary and stir to combine. Bring soup to a simmer, stirring occasionally. Reduce heat to medium-low and simmer for 10 minutes, stirring occasionally, until the pasta is al dente.
Stir in the spinach and cook for 1-2 minutes until it is bright green and wilted. Season with salt and black pepper to taste (if needed). Serve warm.