I suspect that the door isn't so much wood as particle board with a veneer over it and what is disintegrating is the particle board substrate. Frankly, there isn't going to be an easy fix and replacement of the entire door is probably the best course of action. Fortunately there are plenty of places on the Internet that sell cabinets doors in custom sizes. Or you could try to find a local woodworker to fabricate one. It looks to be a simple slab with a rotary sawn oak veneer. You'll need to stain/finish it to match the existing doors. It might make sense to order two doors so you'll have a second to practice your finishing.

I found a manufacturer with solid plywood flush doors with the lipped edge molding. I think this is better than the particle board thing you have.

http://www.thewoodworksinc.com/cabinet-doors/plywood_slab_door_replace.php

Casey

Thank you both for your helpful responses!