Your electrician should do this. When you call about the job, ask if this is something he/she can do.

Normally you wouldn't need a general contractor - however, make sure your electrician and yourself have a clear understanding of how much "damage" he will be doing in the process, how much damage you will find acceptable (i.e. none) and who takes care of any drywall patching/painting required.

A few electricians are used to assuming someone else is coming to fix cosmetic things after they're done ;).

The recess hole is probably much larger than the flush mount box hole. You'll need to deal with drywall repair. I don't know any electrician that will do drywall.

Actually I know several who do drywall repairs of this nature. This should be part of the discussion when hiring the electrician.