It's entirely possible that a header does span across the entire opening. Possible that the house originally had sidelights that were removed (or covered up). For reasons of energy efficiency or concerns about security and/ or privacy. Or the door was being replaced and the owners didn't want to shell out for a door with $idelights.

One suggestion for a quick and relatively inexpensive way to improve the looks would be to remove the applied molding and install raised panel shutters on either side of the door. Various widths are available; the height should be standard. You'll be able to get close enough, I think, without having to resort to a full custom wood shutter.