There are a few free apps that will teach it, I have used the “Advanced Buteyko” ios app.

If you demand extensive peer reviewed medical evidence of some specific quantified outcome before doing any activity in life you will miss quite a lot of valuable things that can’t be easily quantified or measured, or funded academically. There is however actually a lot of medical research on breathwork like this, they just will use the technical terms for what you are actually doing instead of a name like Buteyko.

A lot of the Buteyko studies suffer from small sample sizes unfortunately. I have heard good things about Buteyko from athletes but I’m not well-read on this. I think myofunctional therapy has more Western research done on it and is strongly related