Starter makes 1 second noise at start up when it is cold outside, but it is fine when it is warm. I had same problem last winter and then it started again last week since it got much colder out now.
It is a weird sound. I don't know if I would call it a purring sound, but it does sound like you are hearing the starter. I guess in a way it sounds like a high pitched horse.
I think they cost around $400-500 to replace. It is relatively easy to replace yourself if you know how to open the top engine compartment and the front compartment to the engine directly behind the lower seats behind the carpeted wall. You will need to unplug battery, which means when you reinstall battery, you will need the radio code for your radio if you want to listen to music. If you are lucky, the radio code is written in your owners manual. I guess one of the previous owners or mechanics wrote the 4 digit code down in my manual.
If you fix it at Porsche dealer, they will automatically dial the radio code in for you at no extra charge. But if you will have to call them and ask for the code, they usually charge you for it.