Facial reconstruction surgery. Are there specialized surgeons who do just reconstruction?

Yes . Some surgeons limit their practice to reconstructive procedures.
Yes. Some facial plastic surgeons do only reconstruction. Go to aafprs.Org to find a surgeon.
Plastic Surgeon. Plastic surgery is actually reconstructive surgery. However, big bucks come to play with cosmetic work, so some do only cosmetic surgery - benefits are cash up front and you don't have to deal with insurance companies. Maxillofacial surgeons also can do reconstructive things with the face. You might want to talk to one or two of each to make sure you find what you are looking for.