I've seen quite a few employers who have health insurance that covers stress leave. You might qualify, since what you're experiencing is often the result of long term chronic stress.
How amenable your employer would be to this is another question - if they've seen what you're capable of,and they know how difficult it is to find a capable developer and bring them up to speed on the company's product, they might be open to giving you a month to recharge.
Even if it's not covered by insurance, perhaps an unpaid month off is something you could negotiate? Again, it's very dependent on the employer. I've seen it work, though - and in most cases, the employee almost didn't ask because they'd convinced themselves it would be impossible, but the employer didn't see it as a big deal.
I've been in the position you're in - where you're just staring at the screen, unsure of what to do next. I found it helpful to take steps to restore my creative abilities - I'd lost touch with them after spending years just focusing on code. Being great at writing code is useful when you know what to do, but all the programming skill in the world doesn't help when you can't envision what to do next.
I found The Artist's Way[1] helpful here. Some people like it, others don't. I suggest reading some reviews on Goodreads ahead of time to develop an understanding of whether or not it would be a good book for you. A newer book along the same lines with a slightly different approach is Let the Elephants Run[2].
At the end of the day, your approach to getting past burnout is going to be a personal one. Things that have worked for others may or may not work for you. Keep in mind that many of us have been where you are, and we've been able to work through it and become productive again. You can (and will), too.
Just remember that you didn't get into your current state over the course of a few days; it's been a long, slow grind that has worn you down over the course of years. That doesn't mean it'll take you years to get better, but it does probably mean it would be helpful to treat this as a recovery process that will start small and show progressive gains over time.
(Amazon links, but no affiliate codes or anything dodgy like that (: ) [1]https://www.amazon.com/Artists-Way-25th-Anniversary-ebook/dp... [2]https://www.amazon.com/Let-Elephants-Run-Creativity-Everythi...
With reasons ranging from too risky in terms of expected cost/benefit, to you can't learn as quickly, to family obligations won't give you the time or energy required.
I personally think the learning speed one is bunk -- even if your brain slows down a little (debatable), it seems more than made up for by experience in how you learn most efficiently, on top of general life experience.
So lists like this help remind you that you still can, if you're so inclined. For more inspiration in the creative direction (e.g. writing like in this list), see particularly the well-known book The Artist's Way by Julia Cameron. [1]
[1] https://www.amazon.com/Artists-Way-25th-Anniversary/dp/01431...