So in the meantime, I'm kind of just putting in the minimal effort at work, not doing any side projects, and going on lots of really really long walks. That seems to be slowly helping.
I have read more about being able to use short and long term disability, which might help, or be a viable part of a strategy. If you do this, I'd recommend having your own insurance and not using your state or companies disability insurance.
What I found helps is spending as much time outdoor as possible, preferably in the sun and in the wild nature, especially if they're light hike. Try to do that at least half a day every weekend.
Hope you get better.
I did once have an experience with nitrous oxide where I heard something similar and subsequently had a pretty major shift in my thinking process for awhile.
It has been a little over 2 years since I left my job from burnout, now I am doing something completely different and a lot less intense. But I don't think I could ever go back to what I once did. I have permanently blowen out half my cylinders. ;)
The key distinguishing what I felt with burnout from just being mentally tired is that no problem, no matter how hard it was to solve (or even if I failed miserably at solving it) has ever caused me to instanteously feel 100% mentally exhausted. By contrast, when burnt out, even fairly trivial work tasks would trigger this.