Install Unreal Engine - AchetaGames/Epic-Asset-Manager GitHub Wiki
To install Unreal Engine using the Epic Asset Manager you first need to setup your GitHub Token in Preferences with the read:packages scope

After this is done you need to restart the Epic Asset Manager(#88).
You might also want to set up "Temporary Download Directory"(Needs to fit the engine download) and "Unreal Engine Directories" to specify the place where the Engine will be extracted to.
Because of flatpak sandboxing you may encounter an error when choosing an engine install directory. You need to run this command to overcome this issue
flatpak override io.github.achetagames.epic_asset_manager --filesystem=<path_here>
On the engine tab press the big "+" icon to open the side panel

Here, you will be able to chose the version and if Additional Content needs to be installed.

Wait for the engine download to finish

The engine should show up in the Engines list