Build the broccoli broth: Add the broccoli stems and florets to a large soup pot. Pour in the chicken broth and water, then add the smashed garlic, kosher salt, and freshly ground black pepper.
Bring the pot to a boil over medium-high heat, then lower to a gentle simmer. Cover and cook for about 30 minutes to infuse the broth with broccoli and garlic flavor.
Add the orzo: Stir in the orzo or ditalini pasta and cook uncovered for about 15 minutes, stirring often so the pasta does not stick to the bottom of the pot.
Add the chicken: When the pasta is nearly tender, add the shredded cooked chicken to the pot and let it heat through for a few minutes.
Add the spinach: Stir in the baby spinach and let it wilt into the soup, about 1 minute. Taste and adjust the seasoning with extra salt, pepper, or nutmeg as needed.
Serve: Spoon the soup into bowls and top each serving with a generous handful of finely grated Parmesan, a drizzle of extra virgin olive oil, and extra black pepper if desired.
