Heat the oven to 400°F. Spread the butter in a 13 x 9 x 2-inch baking dish. Add the squash, potatoes, celery root and leeks to the prepared dish.
Heat the broth, cream, thyme and nutmeg in a 2-quart saucepan over medium heat to a boil. Season to taste.
Pour the broth mixture over the vegetables and toss to coat.
Bake for 25 minutes. Reduce the temperature to 350°F and bake for 40 minutes more, or until golden brown and the vegetables are tender. (If the vegetables are browning too fast in the first 25 minutes, cover the dish loosely with foil.)
Sprinkle with the cheese. Let stand for 10 minutes.
*Time-Saving Tip: Use a Japanese mandoline to slice the potatoes to an 1/8-inch thickness.