Place the whole squash in a large pot and add water to cover. Bring to a boil over high heat, reduce the heat to medium-low and simmer, uncovered, until the squash can be easily pierced with a knife, about 45 minutes.
Meanwhile, in a saucepan over medium-high heat, melt the butter and cook it until it turns brown and just begins to smoke, 3 to 4 minutes. Remove immediately from the heat and stir in the nutmeg.
When the squash is done, drain and set aside until cool enough to handle. Cut the squash in half lengthwise and, using a fork, scrape out the seeds and discard. Place the squash halves, cut sides up, on a serving platter. Using the fork, scrape the flesh free of the skin, carefully separating it into the spaghetti-like strands that it naturally forms. Leave the strands mounded in the squash halves. If the butter has cooled, place over medium heat until hot.
To serve, drizzle the butter evenly over the squash. Sprinkle with the cheese and season with salt and pepper. Serve immediately.