How do I put my device in supervised mode from Windows/Linux? I'd like to configure an always-on VPN on my iPhone and as always Apple's absurdity gets in the way. Specifically, they require to put the device in supervised mode, which requires Apple Configurator running on a Mac. I don't have any and I do not plan to buy one in the near future.

I've searched around and it seems like some MDM services allow provisioning of the devices from Windows, which means the supervision process of Apple Configurator has been reverse-engineered.

Now, I'd like to know if there's other Windows or Linux software that allow switching devices into supervised mode, or at the very least some documentation about the process so I can write my own code to do it.
Since you mentioned GroundControl in your link, I would like to make it clear that GroundControl is not an MDM. It is kind of an alternative for Apple configurator. You can supervise many devices at once and also it can be done from a Windows device but physical connection is required. 
Another way to supervise device is by Apple Device Enrollment Program (DEP). Any device which is purchased directly from Apple or from an authorised re-seller can be supervised through DEP. DEP is the only available option for wireless supervision. DEP requires enrollment in an MDM solution. The MDM will assist in additional device management operations. To know more please check out Device Enrollment Program and iOS supervised mode.