Script compatible with distributions based on Debian
This sequence of commands will download the necessary files and execute them to generate the Unofficial Edge Browser AppImage, then you can access the buildMicrosoftEdge / out folder and click on the generated appimage.
Now that there are three or so versions of Edge in the Microsoft download directories, the script downloads all of them and then builds the appimage with the earliest version. However, you can simply delete all but the latest version in the download directory (look for the *.deb files) while the latest is downloading, then the script uses the first one it finds (the latest, or whichever you leave there).