Make the spaetzle dough by whisking together the flour, salt, pepper and nutmeg in a large bowl. In a separate small bowl, beat the pumpkin puree, eggs and buttermilk until well combined. Add the egg mixture to the flour, and using a wooden spoon or spatula, gently stir together until you have a sticky dough. Depending on your puree and eggs, you may find that your dough is too thick. If that is the case, add a little more milk or water, about a tablespoon at a time, until the sticky dough is on the thin side, almost like a thick pancake batter. Let the dough sit in the refrigerator while you bring some water to a boil. This can even be done the night before.
Bring a pot of salted water to a boil. Using a spaetzle maker or colander, pass the dough through the holes and drop it into the water. The dumplings will cook quickly and float to the top. Use a slotted spoon to retrieve the spaetzle. Set aside in a colander to drain.
Make the browned butter sauce by melting the butter in a saute pan over medium heat. Add the garlic and cook the sauce, stirring, for about 5-7 minutes, until the butter is golden brown. Skim any foam from the sauce and discard. Stir in the spaetzle and parsley, and season with salt and pepper. Toss in a generous sprinkling of parmesan cheese and transfer to a serving dish. Serve while hot with shavings of parmesan cheese.
Spaetzle adapted from Fresh Herb Spaetzle by Dorie Greenspan, Bon Appetit (March, 2006), via Epicurious.