Combine the broccoli stems and florets, chicken stock, water, smashed garlic cloves, kosher salt, and freshly ground black pepper in a large soup pot or Dutch oven. Bring the mixture to a boil over high heat, then reduce the heat to low, cover the pot, and simmer for 30 minutes. This will infuse the broth with the flavors of broccoli and garlic.
After simmering, use a potato masher to smash the broccoli directly in the pot. Alternatively, you can use an immersion blender, but be careful not to blend it too smooth; you want to keep some larger and smaller pieces for a nice texture in the soup. I like to leave plenty of chunky broccoli for heartiness.
Stir in the orzo or ditalini pasta to the soup. Cook uncovered for 15 minutes, stirring often to prevent the pasta from sticking to the bottom. The pasta will absorb some of the flavorful broth and thicken the soup.
Add ground nutmeg (if using), and taste the soup for seasoning, adjusting the salt and pepper as needed. Ladle the hot soup into bowls and serve topped with plenty of finely grated parmesan cheese, a drizzle of extra virgin olive oil, and more freshly ground black pepper if desired. For an extra touch of richness, I like to finish with a swirl of good-quality olive oil.