This actually did happen! The Taoist monk is Qiu Chuji (丘處機) and traveled from Shandong through now-Beijing before crossing west into Central Asia and meeting Genghis Khan during his campaign against Khwarezm. A Khitan minister named Yelü Chucai (耶律楚材) compiled the conversations into a document for posterity and guiding future Mongol rulers, known as 玄風慶會錄 (text provided, in case you know Chinese) in Chinese. I have never found an English translation though, and it is written in Classical Chinese, so I won't be able to translate it properly this late at night. However, the main premise of their conversations can be summarized as Qiu Chuji dispelling the idea of eternal life, but encouraging Genghis Khan to follow the principles of wuwei (無爲) and avoid acts of waste and excess in his personal life to cultivate longevity and health. This may have impacted Genghis Khan's policies in his final years and reduced "wasteful and excessive" policies, such as massacring resisting cities and enslaving enemy civilians

If I may add onto this question with one of my own -

I did some light Google searching and found that Ghengis was interested in, among other things, the 'philosopher's stone'.

What would the Eastern Chinese understanding of the Philosopher's Stone have been at the time? Would this understanding have been based on earlier Iranian sources? Were there any prominent scholars from eastern China who wrote about the Magnum Opus, of which I may track down a translation?

Thanks in advance, sorry if tacking a question on breaks the rules.