Meanwhile, in a large saucepan or Dutch oven, heat oil over medium heat. Add onion and cook, stirring frequently, until tender, about 5 minutes. Add garlic and mushrooms and cook, stirring occasionally, until mushrooms are tender, about 5 minutes. Add broth and rosemary; bring to a boil.
Stir cooked rice and chickpeas into the broth; return to a boil. Reduce to a simmer, cover, and continue cooking for 5 more minutes to allow flavors to blend.
Stir in spinach and cook, uncovered, until just wilted, about 1 minute. Season with salt and pepper. Garnish with cheese; serve immediately.