In a pot, heat the olive oil over medium heat. Sauté the onion, celery, and carrots for about 4 minutes. Pour in the broth, thyme, and chicken, season with salt and pepper, cover with a lid and bring to a simmer until the chicken reaches an internal temperature of 160°F using a thermometer.
Remove the chicken onto a cutting board. Add the orzo into the soup, cover with a lid and return to a simmer. Using two forks pulling in opposite directions, shred the chicken apart then return it to the simmering soup. Simmer until the orzo is tender (refer to the instructions on the package for cooking time). Stir in the spinach until wilted.
Taste and adjust the seasoning with salt if needed. Remove the thyme sprigs. Garnish with parmesan.