How can I remove a large screw from a door, when the nut is broken off?

I'm trying to replace handles on old wooden doors, that are screwed on with long screws. The screws seem to have been put in by machine or something and go right through the doors from the back to the front, they have no nut on the inside end so there's nothing to grip with a spanner or screwdriver. The doors are solid wood, I don't want to drill new holes. I've tried pliers but this only seems to do damage to the thread. Thanks!