I am assuming that you already connect your laptop and work PC to the same network, but realize that is uncommon in serious work environments. Otherwise, skip this whole thing!
Google led me to only to pay-for options describing what you sort-of want. JamCast seems to reroute audio but I thought it was free till 5 minutes before ending this post. Sorry. I found it's only a demo with a pay-for conversion. That said: Jamcast or anything else, really, needs to be installed on both PC's, and you have no choice but to convince your boss to install something because Windows doesn't just let us squeeze its audio through its network card like Linux supposedly can. You will also need to install it on the headphone PC (easy, it's yours, right?)
Besides the job permissions hurdle, it's unlikely that the transmitter nor receiver program will start automatically without your clicking. You have to ask around to find your transmitter's IP address if you're not allowed to peek. A Startup shortcut that opens the program may be needed on both PC's. That's for the "forget it" part.
As an "instead of" last resort, if your PCs have no firewall blocking and it's legal to connect them, try
mstsc.exe type.your.IPaddress.inhere (from your Start Menu \ Run). It remote-controls it and redirects all audio when doing so. It is over kill and inevitably does password-lock the transmitter PC. Your screen then will show the real session on the receiver PC, but your boss will wonder why his PC shows only a login screen instead of a working e-mailer. Your options are limited, and I am too tired to try reading the Wiki on what looks like a Pro standard for audio handling