I imagine most employers would balk at such a reversal of norms, not to mention the blow to their egos. That said, a lot of employers are doing outreach campaigns that are basically the equivalent of what you’re asking about. Your idea is funnier, though.
@Dutchess_III Since we’re in a labor shortage, potential employees can be picky about who they work for. So the question is about turning the tables and making the boss submit letters of reference (i.e., “give me three references who will tell me why I should work for you instead of someone else”).