Is a genioplasty done by a jaw surgeon? Would insurance cover it?

I had braces for a bad overbite in my 20s and although braces improved the position of my teeth, the overbite left me with a recessed chin. I want to fix it, but am confused about what type of surgeon would fix a recessed chin. Do I need to go back to my orthodontist first? Or is it purely a cosmetic issue? I can pay privately, but would prefer to save thousands if my insurance covered it. :)

For reference, my top teeth are still a good bit forward when I close my mouth. It's just that I no longer have horrible buck teeth.