A Shield is pretty pricey for doing this. Steam Link is much cheaper, and is on sale for $5 right now. (I believe shipping is free on GameSpot.)
https
/www.gamestop.com/pc/accessories/steam-link/121866
http

/store.steampowered.com/app/353380/Steam_Link/
The way it works is it uses the h.264 encoding hardware on the GPU to turn the game video into a h.264 video stream in real-time. The Steam Link has h.264 decoding hardware on it (as does pretty much every smartphone nowadays), and thinks it's just playing a streamed video. There's some slight pixelation on fast action scenes. And due to the real-time nature (can't buffer like Netflix does) it works more smoothly over Ethernet than over WiFi.
The Steam Link app would do the...