I don't know. I am sure that it will come to you once you take your focus off of "wanting" to do something to help. Funny how that shows up.
The actor, James Earl Jones, who was/is the voice of Darth Vadar on Star Wars is a stutterer. He said that he had his greatest trouble when he is trying to speak without first knowing what he was going to say. Which, I guess, is why being an actor is so suited to him.
So, like you said, when your professor is speaking about that which he knows very well, he doesn't have as much trouble as when he is addressing that which is ?boring? to him.
He obviously knows his stuff, but maybe a prepared speech would be handy for him?
I have also heard of people who have cured themselves of stuttering, but I don't recall what they did to do it.