I got a metal box with HDMI and an ethernet ports which encodes the HDMI video + audio and serves/publishes H.264 RTMP/TS segments. My intended use case was actually working with Pi's on live streams, installing fresh systems and the like where encoding might not be available anyway.

I couldn't find the one online but it looks like there are a few out there. Side note/tangent, the Pi3 was powerful enough to transcode 720p down to lower resolutions in real time.