What podcast were you listening to?
I know a little about Beaumont as I'm a big Twilight Zone fan, and a couple years ago I watched every episode again with my two sons (who were seeing it for the first time). I read a collection of Beaumont's stories a while ago, too. Pretty good short story writer, although I think he was even better as a screenwriter. The episode you're referring to above was one of his best, "Long Live Walter Jameson." The great Kevin McCarthy (star of the original Invasion of the Body Snatchers) played the ageless man.