Found a fix that worked for me!
Simply turning off “Scan my library automatically” fixed the problem for me. The box needs to be un-checked. I know this is a nice option to have, but if you have a large library, EAC3 Audio will not transcode correctly with “Scan my library automatically” turned on (checked). You will need to manually scan your library when you add new files, or you can set “Scan my library periodically” to 15 minutes, 30 minutes, 1 hour, etc.
Where is “Scan my library automatically” located? Open your web browser and open Plex (https://app.plex.tv/). Click the WRENCH icon (top right) or click on your “Account Settings” (top right). This should open a bunch of settings on the left of your screen. Click on “Library”. The top item on the right-hand side of your screen should be “Scan my library automatically”. Make sure this option is un-checked, then click “SAVE CHANGES” (bottom of screen). You also may need to stop Plex on your NAS and then re-start it after you save this un-checked this option, but probably not.
Why does “Scan my library automatically” affect EAC3? From what I understand, Plex uses the “inotify” Linux command to process EAC3. If you have a large library, the “inotify” command will reach its limit by watching at all your files, directories, and sub-directories, which will cause EAC3 to hang or fail. It may seem odd to use the “inotify” command for EAC3, but apparently Plex has a reason for doing so.