In a large saucepan, bring the water to a rapid boil. Stir in brown rice and return to a boil; reduce heat to simmer. Cover and cook until rice is tender and has absorbed all the liquid, 30 to 40 minutes.
Meanwhile, in a large saucepan or Dutch oven, heat oil over medium heat. Cook onions, stirring occasionally, until tender, about 5 minutes. Add garlic and mushrooms; cook, stirring occasionally, until mushrooms are tender, about 5 minutes. Add broth and rosemary; bring to a boil. Cover and remove from heat.
Stir rice and chickpeas into broth mixture; return to a boil. Reduce to a simmer, cover, and continue cooking 5 minutes more to allow flavors to blend.
Stir in spinach and cook, uncovered, until just wilted, about 1 minute. Add 1/2 teaspoon salt, to taste, and season with pepper. Serve immediately, sprinkled with grated Parmesan and drizzled with more oil.