In my experience they don't have to have a part of the mouth, but that definately helps them to recover faster. You can also just cut the cap off from the stalk. Leave the caps in coral rubble to attach, then superglue to your rock.
The stalks should also regrow into new shrooms. You'll see a lot of stringing white stuff, don't worry it all gets sucked back in. I usually use scissors but calfo suggested a sharp blade. Trick is to get them in one slice before they contract. Depending on how long the stalks are it may be hard to cleanly cut the cap off. I also find that the stalks heal faster with part of the cap left on it. So ideally cut off only 2/3 of the cap, which can be hard depending on how big the shrooms are.
There are many ways to do it, and you'll probably find it very difficult to make accurate cuts, but don't worry they're quite resilient, and if one half doesn't make it, usually the other half does.