Who did ? I can't find any lore info on uespwiki apart the stuff relative to the game .

High Hrothgar? I would imagine the first Greybeards did. One of stone tablets on the way up the mountain says:

Jurgen Windcaller chose silence and returned

http://www.uesp.net/wiki/Lore:Five_Songs_of_King_Wulfharth, specifically "Kyne's Son," claims that Wulfharth rebuilt the 418th step "of" (not "to," although I'm not sure that that really matters, since the mountain is sometimes identified by the same name as the monastery) High Hrothgar. This indicates that the monastery was built long before the Battle of Red Mountain and the foundation of the Greybeards, at least as we know them. I personally believe that the Throat of the World was so important to Nords (being the place where they were breathed onto the world by Kyne) that they had already built a monastery or something up there that the Greybeards later took over--and might well have expanded or rebuilt--for their own use.