If you restored a backup over the top of an existing system then files with
the same name were overwritten with the original files. In this case it is
very unlikely that the overwritten files could be recovered. Unformatting is
possible where files have been deleted by formatting the disk, which is a
very different issue to overwriting files with older copies. Unformatting
would not help your situation.

If you restored using some sort of disk image, then it is possible that some
of the more sophisticated data recovery utilities would be able to access
parts of the disk not yet written to, and recover the contents. You could
investigate the professional recovery services, but they would require a
complete description of the software you used and the procedure that was
involved.