OK so here are several issues you need to solve...
1) You need connectivity so that Farnsworth A can connect to Farnswroth B. That means WIFI or Cell service. If it's cell, you might as well just put an iPhone inside a Farnsworth case and use Facetime, but I'm assuming that's not what you want to do.
2) You need a screen. You will not find a round screen device, but you can get flat LCD-types. They will have to be fairly big in order to cover the whole circle.
3) You need a way for each Farnsworth to identify each other so you know who to connect with and what their address is, whether phone number or internet address.
4) You need a little camera - easy enough to get.
5) You need a protocol for sending two way video and audio. You do not want to try and make this yourself.
6) You need two of them so you're not talking to yourself
I'd personally look into how I could make a miniature Skype platform, and just utilize it. You do so not want to reinvent the wheel of video conferencing, but rather leverage something that already exists.
As to Raspberry pie? Sure maybe - can you run Skype on it and run a low power large flat screen? Then maybe you got something if you're willing to work at it.