How do I restore an iPhone from a backup located on an external drive?
I was unable to restore from this backup because I had symlinked the iPhone backup's individual folder,

Volumes/External Drive/iPhone Backups/UF34...


into my Backups folder located at 

~/Library/Application Support/MobileSync/Backups/.


This causes some issues. What you need to do is to symlink the backup folder into the MobileSync directory in your user folder. In other words, you need to create a symlink for the folder which contains all your iPhone backups (the folders with the weird numbers),

Volumes/External Drive/iPhone Backups/Backups


inside this folder on your disk

~/Library/Application Support/MobileSync/


I believe you need to rename (or delete) the Backup directory which is already on your hard disk if you want to do this.

I realize that this process might be slightly confusing if one has not used symlinks very often, so feel free to leave a comment here and I will try to make this answer clearer if I can.