Thanks guys. It seems that something very similar to jack-nuts were used originally, only the threaded section on the original part is a little longer than in your pics.
When reinstalling the mirror and tightening the bolts into place, part of the threaded section broke loose and when the bolt was backed out, that part came out with it. Since part of the threaded section remained in the door, I was able just to buy a slightly longer bolt and that is holding well for now. In all likelihood, if I ever back that new bolt out, I'll be installing new jack-nuts shortly thereafter, so this will definitely be useful info. Thanks again.