Pour the broth into a 3-quart / 2.8 L saucepan (20 cm / 8-inch) and add the Parmigiano rind if using. Bring it to a bare simmer over medium-high heat, then keep it hot on low while you start the soup.
Heat the olive oil in a 4-quart / 3.8 L heavy-bottomed pot (24 cm / 9.5-inch) over medium heat. Add the onion, celery, carrot, sage and a pinch of the salt. Cook for 6 minutes, stirring often, until the vegetables are soft and translucent but not browned.
Add the pumpkin cubes and stir for 2 minutes to coat them in the soffritto. Pour in the hot broth and rind, bring back to a simmer, and cook partly covered for 10 minutes, until a cube crushes easily against the pot.
Mash about half the pumpkin against the side of the pot with a potato masher or the back of a wooden spoon, leaving the rest in chunks. The broth should turn pale orange and slightly thick.
Stir in the rice and simmer gently, uncovered, for 15 to 17 minutes. Stir every few minutes and scrape the bottom so nothing sticks. The rice is ready when it's tender with a slight bite at the center and the soup looks loose, like thin porridge.
Take the pot off the heat and remove the rind and sage leaves. Stir in the butter, grated Parmigiano and nutmeg, then season with the remaining salt and black pepper. Cover and rest for 5 minutes.
Ladle into warm bowls and add a splash of hot broth if the soup has tightened. Top with olive oil, pepper and extra Parmigiano, and serve within 10 minutes.