The NES Classic is hard to find. The Nintendo Switch is hard to find. Should you be lucky enough to have both and somehow want to pair a Switch controller with your NES Classic, it's possible. You just need the help of a little dongle.
8Bitdo's Retro Receiver, which sells for about $16 on Amazon (roughly £13 or AU$21), works with PS4/PS3 controllers, Wii Remotes, or Wii U Pro controllers. A new firmware update posted by 8Bitdo allows Joy-Cons or even the Switch Pro Controller to be paired to the NES Classic, too.
The firmware update gets installed via a download to your PC or Mac and then connecting an included USB-to-micro-USB dongle into the wireless receiver.
Both the Joy-Con and Pro controllers cost far more than a typical wireless NES Classic controller, but if you have a Joy-Con handy (or one of 8Bitdo's wireless controllers that include the Retro Receiver), maybe it makes sense.
Yep, it works
I tried it out with firmware sent by 8Bitdo, and it really works. It seems intimidating at first, but it's actually easy: download, plug the Retro Receiver into your computer and install. After that, all you have to do is press and hold the small pairing button on the receiver once it's plugged into NES Classic, and press and hold the pairing button on a Joy-Con on a Switch Pro controller. And you're done.
The Joy-Con works pretty nicely as a tiny NES controller, minus the d-pad. (The analog stick stands in as best it can.) Buttons are mapped properly, with "start" and "select" moved to the tiny SL and SR buttons on top of the Joy-Con.
Reconnecting Joy-Con controllers with the Switch is easy too. Just slide the Joy-Con right back into the Switch. But then you'll need to pair the Joy-Con with the Retro Receiver again.