more info here: https://www.bootladder.com/2018/12/20/chromecast-1080p-as-external-display.html
With a Chrome Cast and a Linux host, it only mirrors a tab or a desktop.
It can't extend a desktop.
Also, you can't get resolutions higher than your host's resolution.
But here's a workaround, where another X server is used to run Chrome,
and so you can get the other desktop on the Chrome Cast.
The last thing piece is to create a virtual display using xrandr for that X server,
so you can use arandr to allow your mouse to go between the 2.
A stab at automating it looks like this; missing piece is that the ChromeCast should be configured by a command line.
#!/bin/bash
Xephyr -br -ac -noreset -screen 1920x1080 :1 &
sleep 1
DISPLAY=:1 awesome &
sleep 1
DISPLAY=:1 chromium-browser &
Of course set your display number ie. :1 and your window manager accordingly.