Preheat oven to 350 degrees F. Place rice in a shallow baking pan and toast, stirring occasionally, until golden brown, 8 to 10 minutes.
In a large soup pot, heat olive oil over medium-high heat. Add onion and cook, stirring frequently, until golden, about 5 minutes. Stir in garlic, salt, nutmeg, cayenne, vegetable broth, and rice and bring to a boil. Reduce heat to low, cover, and simmer until rice is tender, about 12 minutes.
Stir in spinach, Parmesan cheese, and lemon zest, simmer until spinach is wilted, about 3 minutes longer, and serve.