Although it it not precisely "one" click you can easily use the "Next up" list to pretty much accomplish what you want. It is what I do to be able to keep track of several series I am watching.
Also on a number of clients I "think" if you simply press "Play" on a series itself playback will resume where you left off. I am not sure about that as I have never tried it but I think I remember reading that it does work.
True! Many video players like VLC and Media Player Classic has this function to continue where I left off, but I want to use Emby's interface for the reasons I said up there. Even if I use it on my PC browser. So much more easy and elegant.
The downside, for me anyway, to use video players, is to navigate trough the folders to find the last episode. I use video player a lot, so the Last Played playlist is always all messed up.
You also could configure your home screen to put continue watching right at the top.
Luke, can't thank you enough. It's so simple. Next time I'll take the dive into the Emby's settings. Thank you.