So the door would not open eh??
Good thing no one was around....hehe.
Well, cutting the door is out. They are usually full of foam and have the structure parts located where you need to cut it out.
The easier solution is to remove the entire door unit, jamb and all and raise it the inch you need. This will require some modification to the siding outside.
Let us know what type of siding you have and we can go from there.