I came across this place when researching our 2010 trip to Italy with two daughters. It did not work for us at that time, but I have always longed to stay at a place like this. Of course I lost track of it, forgot the name, and could not find it again with any kind of search.
Then today I came across an article about "alberghi diffusion" in Italy and tried googling that term. And there, among the alberghi diffusi in Tuscany, it was. Castello di Gargonza is the name.
Articles on the alberghi diffusi:
http://www.travelandleisure.com/articles/reviving-the-italian-village
https://ecobnb.com/blog/2013/11/alberghi-dffusi-hotels-italy/