Since no one else has answered, I'll give it a shot. Whiskers do serve a purpose. They help with balance and chinchillas can "see" more of their environment using their whiskers that relay information to their brain. That being said, they do lose whiskers as they grow new ones in like human hair.
So what *I* would do first is watch your chin. Does the hair seem to be bothering him? Is his balance off? Any other changes in behavior?? If the answer to those 3 are no, I'd probably adopt a wait-and-see attitude, keeping a eye out for any of those 3. If it isn't bugging him, it shouldn't bug you and it will eventually grow out.
If the answer to any of those is yes, then personally, I'd take a pair of blunt-nosed scissors and cut that one whisker just behind the hair stuck to it, leaving as much of his whisker as you can. That's me though, I'm comfortable doing those kinds of things. I don't think it's a big issue really *shrug*
Hope that helps!!
~ Tina