There are various approaches here; for your specific use case, you require a solution that can output through both headphones and desktop speakers (monitors).
Option #1, which is likely the option you're currently utilizing, is to plug the headphones directly into the 3.5mm out on your motherboard or case, and the Yamaha HS8 directly, as well, through 1/4" TRS to 3.5mm adapter. After that, you would need an output application to play the music, such as iTunes, Roon, Tidal app, etc. and you are set. The DAC is performed by the motherboard internal soundcard, the internal amp powers the cans, and the powered monitors receive output from the internal sound card and are self-amplified.
For 99% of people, the above is sufficient and will provide a listening experience that is enjoyable with limited audible differences than other types of set ups.
As you have come into more funds and have the desire to upgrade, you have two other options for your use case; these options are way more fun to put together, even if audible differences are typically quite nuanced.
2) An entirely Integrated solution (all-in-one DAC, preamp, headphone amp and speaker output); as you are familiar with Schiit brand, I would direct you to their Asgard line for 200 bucks. This provides a DAC/headphone amp in one box, as well as pre-out which could feed your Yamaha powered monitors via a RCA to TRS adapter/wire. Very simple, cheap, and can provide you with an audible upgrade.
3) Separate components (this can be completely separate DAC, preamp, head amp, etc. or it can be separated DAC and preamp/head-amp; all depends on how much fun and funds you want to have and have); sticking with Schiit, this would be getting a Modi DAC feeding a Magni head amp/preamp out to power monitors, or you can even break it out further. The source here, ultimately, is still your PC feeding the DAC via an output (USB is my preferred).
I used the cheapest Schiit components above as an example, but this can all tremendously scale. However, basic process is the same.
Hope this helps.