With a fork or sharp knife, poke holes all over squash. Microwave it on high 10 minutes. Let sit until cool enough to handle. Halve squash lengthwise; scoop out and discard seeds. Flesh should be tender when scraped with a fork; if it isn't, microwave on high (cut side down) in 1-minute intervals until tender. Let sit until cool enough to handle. Scrape out flesh (discard peel) and mash until smooth.
In a large bowl, thoroughly combine 2 cups mashed squash (save any extra for another use), 1/2 tsp. salt, the white pepper, and nutmeg. Stir in flour, 1 cup at a time, until a dough forms (it will pull away from inside of bowl).
Turn dough out on a generously floured work surface. With well-floured hands, knead dough 10 to 12 times.
Divide dough in half and cover 1 batch with plastic wrap. Roll other batch into a 3/4-in.-thick rope and cut into 1/2-in.-long pieces. Put pieces on a floured baking sheet and set aside. Repeat with remaining dough.
Bring a large pot of salted water to a boil. Boil gnocchi until they rise to the surface, about 4 minutes; cook 30 seconds longer and then lift with a slotted spoon, making sure water drains from gnocchi, and place in a large serving bowl. Gently toss with butter and top with cheese and a sprinkling of black pepper. Serve hot and pass extra cheese at the table.
Note: Nutritional analysis is per serving.