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Mental health issues are routinely recognized as serious health issues. I work for a major US company and employees are encouraged to take short-term or long-term disability leave, yes even for mental health issues. You are already paying for it. Check your benefits and talk to professionals to make sure it is properly handled.

After confirming your benefits and speaking with healthcare professional, talk to HR before your manager. Your manager may not know all the benefits available or try to hide them from you.

Take as long as you need to recover. You may not have hobbies but sometimes vegging out in front TV for a few days is all you need to recover.

Also forget about all the healthy stuff like exercise, meditation, etc. Just sleep as long as you can. Burning out means your body is under maximum stress. You don't want to add to your plate. Remove things from it.

Relaxing experiences maybe okay like massages, bathes, a slow walk.

Finally, ancedata, I know someone who was getting burned out. They didn't show up for work for a week. When they got back, their boss was furious but they said they were burned out. HR got involved. Boss got in trouble for creating unhealthy work environment. Person was told to take more time off and see a therapist. This was at a major US company.