You already have one PC where Studio running, of course, if you want to show single window and you alt-tabbing a lot during streaming session then Fullscreen Projector is a valid option. Another way - is to capture whole display (cloned by windows) and still use the keys from second PC to fully control the scene.
If you don't have capture card in second PC (or want to remove it) then it is possible to capture content by OBS Studio on gaming PC, do light compression (MPEG2 for example) and transfer the data to second PC via LAN. Then you can import this data stream via VLC Player Source or Media Source in OBS Studio that running by "streaming PC", add scene decoration (if required) and start streaming to the Internet. This is common usage of OBS Studio when application running for both PCs.