Unfortunately I didn't document my own repair, but I'll try my best.
UPDATE: While AKNES are supposedly revising the product - and other users in the thread are exploring less destructive solutions to the problem - I would advise taking a wait and see approach over rushing into following this procedure.
Firstly, I must stress that this is a destructive "fix" that carries a high risk of causing irreparable damage, and should only be attempted if you are prepared to
lose the stick. As mentioned above, the metal casing does not come off easily - if it breaks, it could be impossible to reassemble. Altering the plastic also changes the feel of the stick, taking off too much could make it unpleasantly "wobbly".
Also please, nobody try this just because I suggested it made the stick quieter! There is no guarantee you will be able to completely eradicate the noise to your liking. AKNES seem willing to offer refunds over the issue, so try that first.
With that said, I would strongly advise that the first thing you do is go into the calibration screen and test
where the click is occuring, and decide whether it's truly necessary. For example, my right joycon only clicks/blips as it approaches the outer ring - so although the sound is annoying, the impact on gameplay is negligible. However, the output of my left joycon was visibly jumping ahead before even reaching the halfway point, so for me, that was an unacceptable detriment to gameplay.
You should also make a note of the stick orientation before disassembly, it is important that you remember which direction is which on the stick once it is all in pieces - or you could end up misdiagnosing the issue.
Borrowing Spawn Wave's teardown picture (and the picture AngelofWoe posted earlier, to show the missing piece) the part you will need to alter depends on which direction your stick is clicking in.
View attachment 348793
You should be able to see this movement for yourself if you use the spring to hold everything in place and move the stick:
View attachment 348759
If you take out the affected arm, it looks something like this from the side:
View attachment 349257
(
Remember, if your stick only clicks in one direction while fully assembled, your priority is to reduce the lift that occurs in that direction. To lower the risk of "over-shaving", it would be best to leave the edges affecting the other three directions completely untouched on your first attempt.)
You need to
slightly taper off that top edge on the side that is catching
when you push in the problem direction, just enough that you can see that it has stopped lifting when you test with the spring in place - no more than that. You will need to use your own judgement here, look closely at where your own stick is catching and be sure to
thoroughly test the effect that
even the smallest alteration you make is having. I cannot stress enough that the amount required is
not even enough to be visible, and as I mentioned earlier taking off too much will change the feel of the stick.
Hope that helps, if anyone is determined enough to go through with it.