You would need at least a RX 460 / RX 550 or a GTX 1050 to use hardware encoding.
GT 1030 does not support nvEnc.
If you do decide on using video card, nVidia cards have a slightly better video encoder (as in more quality per bitrate)
Your two E5-2420 are powerful enough to encode 1080p 60fps in software using x264, (as long as you use right quality and speed settings) and you'll get better quality.
You can use NDI to stream the image to the other server and have the server compress it and upload it.
See the video below
Your server can also handle more powerful processors. Your current E5-2420 have a core of ~ 7000
Basically, you'd get up to 45% more performance for around 60$.
There's also E5-2450L (8c/16t, 1.9 Ghz) are out there for around 19$ and have about 8200 score so it's not really a big enough performance bump to be worth it.