Remove the sweet potato skins (you can peel away the skin with your fingers, or slice them in half lengthwise and scoop out the flesh). Use a potato ricer (the ideal scenario if you have one) to mash the flesh into a medium bowl. If you don’t have a potato ricer, use a fork and mash it gently but throughly—try to keep the potato as fluffy as you can. Measure out 1 cup of mashed sweet potato and place it in a large bowl. Measuring is important so that you have the right ratio of sweet potato to flour; if there’s extra, save it for another use*. Let cool to lukewarm.

In a medium bowl, whisk together the Parmesan, flour, and salt. Sprinkle over the sweet potato mixture. Use a fork to gently combine, stopping as soon as no dry bits of flour are visible.

Flour your counter well. Grab a bench scraper if you have one and keep it near by (this will make your life MUCH easer). With your hands, gently form the mixture into rough ball, then transfer it to the counter. Knead gently for 1 minute, until the dough is a sticky ball and a little smoother than before. It will be very sticky and cling to your hands; if it's clinging to the counter, add a little more flour and use a bench scraper to scoop it back together.

Bring a large pot of water for cooking the gnocchi to a boil. While that happens, cut the butter into 4 pieces, then place in a large nonstick skillet and heat over medium-high. Cook, swirling the pan periodically, until butter begins to sputter and foam. Turn off the heat, then whisk up any brown bits that collect on the bottom—this will take about a minute; don't brown the butter completely because you'll be heating it up again. The color may not change much, but it should smell a little nutty. Immediately add the sage, salt, and vinegar.

Let’s get back to the gnocchi—your water should be on its way to boiling. Line a rimmed baking sheet with parchment paper and dust it with flour. Dust more flour onto your counter.

With a bench scraper or butter knife, cut the dough into 4 pieces. Gently roll each piece into a long rope that is 3/4-inch thick, adding more flour to your counter as needed so you can roll it out without it sticking. Be as light with your touch as you can. Cut each rope into 1-inch pieces, then place the pieces on the prepared baking sheet. Repeat with remaining dough.

Once the water is boiling, salt it generously (about 1 tablespoon). Gently drop half of the gnocchi into the boiling water, spacing them out. The gnocchi will sink to the bottom for a minute or so, then float. Let it float for about 30 seconds, then it's ready to be removed.

While the gnocchi boils, return the skillet with the butter sauce to medium-low heat. With a slotted spoon, scoop the cooked gnocchi into the sauce, shaking off excess liquid. Place it in a single layer in the skillet and let it sizzle while you cook the remaining half of the gnocchi. Add the rest of the gnocchi to the skillet. Sizzle a minute or so more, turning the gnocchi so that each piece gets a bit of a sear (adjust the heat as needed). Serve immediately with a sprinkle of additional Parmesan.