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Cook the sweet potatoes: Scrub, dry, and pierce in several places with a fork. Place sweet potatoes on a microwave-safe plate and microwave on full power until completely tender, 8 to 9 minutes. (To roast the potatoes in the oven instead of the microwave, see “tips,” below.) Let cool slightly.
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Slice the sweet potatoes lengthwise to expose the interior. Scoop out the flesh, being careful not to include large pieces of skin mixed. A muffin scoop can be a great tool for scooping.
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Place the sweet potato flesh into a potato ricer with the large hole screen and press through. If you don't have a potato ricer, use a potato masher or fork. Keep in mind you want the potato to be light and fluffy, as opposed to densely mashed. You should have about 1 1/4 cups (225g) prepared sweet potato. (If you have significantly more sweet potato flesh, save it for another use.)
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All the potato to cool at room temperature until it's lukewarm (about 20 to 30 minutes). Add the egg, mixing until well combined.
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In a small bowl, whisk together the Parmesan and flour. With your fingers or a fork, gently work the Parmesan and flour mixture into the potato mixture. Use a light touch to keep the mixture from becoming dense. You're looking to make a soft dough that just holds together. It's OK if there's flour still left in the bowl; you don't want to knead the dough thoroughly.
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Cover the sweet potato gnocchi dough with plastic wrap and allow it to rest while you bring 2 quarts of water to a rolling boil. On another burner, melt 2 tablespoons butter in a large skillet. You'll need both the water and skillet ready to go when you start cooking the gnocchi, as they cook in just a few minutes.
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When the water is nearly boiling, use a bench knife to divide the dough into three pieces. Divide those pieces into smaller pieces, about 1/2" wide and about 1" long.
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Gently roll each gnocchi in flour to prevent sticking. Using a gnocchi board or fork, press grooves into each piece and set it aside as you roll more pieces.
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Place the gnocchi into the boiling water and stir gently. They'll sink at first but will begin to float after a few minutes. Once all of the gnocchi are floating, cook for 1 to 2 minutes.
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Making sure the skillet is hot, use a slotted spoon or strainer to transfer the gnocchi into the hot butter. Fry until they turn golden brown on one side, then flip and cook for another minute or two. Serve the sweet potato gnocchi while it's still hot with additional Parmesan, small pieces of feta, smoked mozzarella, or Gorgonzola cheese; or with your favorite sauce.