While that's happening, halve the onion and peel the papery outer skin off. I personally used to be totally hung up on sweet onions like the seasonally dependent Vidalia and it's winter sisters from southern hemisphere, but, of late, we've really been appreciating our local NY/NJ onions, which are less sweet, but very deep and earthy. So, your call on what kind to use, but make it yellow. Slice each half into strips and toss into either the same skillet or a new one over medium heat with a little olive oil. Allow the onion to caramelize and brown slightly at the edges, stirring as you go and cooking for 5-7 minutes before throwing in the garlic, which should be smashed, peeled, and minced. Cook for 10 or so minutes, allowing the mixture to become fragrant and lowering the heat if either the onion or garlic start to blacken.
Keep an eye on all of that as you start in on the gnocchi. After the sweet potatoes have cooled enough for you to work with them, mash them with a potato masher or a fork in the mixing bowl until you've got a smooth, lump-free consistency. Now add your two flours and the corn starch and mix together until you've got a smooth but stiff dough that peaks a bit when you remove the fork or whisk. For the flours and the starch, we tend to use Bob's Red Mill. If you're gluten-free or just experimenting with lessening your gluten intake, you should definitely check out their extensive line of gluten-free flours. Basically, if you can grind it up in a mill, Bob's sells it.
In another skillet, warm a tablespoon of olive oil over medium heat. Using a spoon or just your hands, begin to roll the gnocchi dough into small dumplings, roughly an inch in length. They should hold together pretty well. If they don't add a little more starch and/or flour and mix thoroughly. Roll enough dumplings to cover the skillet bottom without having them touch and then add them to the warm oil. Allow the gnocchi to brown, carefully flipping to cook on both sides, usually sauteing for about five minutes per side. Remove and set aside as you move on to your next batch, repeating until you're out of dough or have as many dumplings as you want.
Back at the cutting board, take the well-washed Maitake and cut it into chunks of about one or two inches square. When you're cutting the mushroom up, be sure you remove + discard all of the woody material at the base of the stem. We usually get our Maitake from our weekend Greenmarket, which features Orange County, New York's Madura Farms mushroom vendor as a mainstay. So, yes, we're a bit spoiled. But you can find them at higher end grocery stores too if you don't yet have a trusted mushroom supplier, though they tend to cost a lot more and not be nearly as fresh. If you can't track them down, substitute your favorite other mushroom. Don't like mushrooms? Eh...I'm not sure why you read this far. So, add the mushroom to the onions + garlic and increase the heat a bit. Cook uncovered for 10 minutes or so, stirring and allowing the mushroom to reduce and brown, maybe even blacken a little. Once that's happened, add the half cup of a dry white wine—nothing too sweet; maybe a pinot or sauvignon blanc. The heat should be high enough for you to hear an audible sizzle when you add the wine. Lower the heat a little and cook until the wine's reduced to a nice sauce.