I find this option unlikely to exist, because it is an unusual workflow for agile methodologies (the ones that popularized this "post-it" approach to handling issues). If you are working by yourself and just want this functionality for personal purposes, you could use this Chrome extension. It opens the issue form on the same page and allows you to close it without having to leave.
The idea behind the columns is to provide a big picture to all collaborators. Moving an issue to "Done" serves the purpose of telling your workmates that work has been done and, now, they can work on stuff that requires it. It doesn't necessarily mean it has been closed (as in everything has been completely dealt with), only that they don't need to worry about it anymore.