My impatience won, and all I can say is wow! My one other fig on this tree is starting to swell, and you can bet that it is bagged away from the local moochers. To me, the flavor was that of a perfect peach martini. I probably could have left it on for other day, at the top of the thick white part (sorry, I am unfamiliar with the terminology) left something to be desired. I was just afraid that since it had been opened, it might go bad before I picked it.
I'm also attaching a picture of the tree itself, if that can help in identifying it at all. Thanks everyone!