Special equipment: Stand mixer
Preheat the oven to 350 degrees F.
Using a fork, lightly prick holes all over the sweet potatoes. Set the sweet potatoes on a foil-lined roasting tray and bake until fork tender, 30 minutes.
While still hot, cut the sweet potatoes in half and scoop the flesh into the bowl of a stand mixer fitted with the whisk attachment. Add the 1 cup Parmesan, olive oil, brown sugar, thyme, nutmeg, 1 tablespoon salt and 1 teaspoon pepper. Beat on medium speed until the ingredients come together. Increase the speed to high and mix until light and airy, scraping down the sides of the bowl as needed.
Dump the sweet potato mixture onto a lightly floured work surface and gently knead, adding small amounts of flour at a time, until the dough comes together and holds as a single mass. When done, the dough should be light and tender to the touch. Place in a floured bowl, cover and let rest in the refrigerator for 30 minutes.
Divide the dough in half on a lightly floured work surface. Roll each half into a 3/4-inch-thick rope. With a sharp knife, cut the ropes into 1-inch "pillows."
Bring a medium saucepot of salted water to a boil. Add the gnocchi and cook until they float, 2 to 3 minutes.
While the gnocchi are cooking, set a large nonstick saute pan over medium-high heat. Add the butter and melt until frothy and nutty-smelling. Drain the gnocchi, reserving some of the cooking water; add the gnocchi straight to the saute pan. Toss to brown lightly, then season with salt and pepper. Once browned, you can add a little of the pasta water to make a light sauce. Divide among serving plates and shower with the remaining 1/4 cup Parmesan.