Displaying fields from an entity reference of another entity's field
You can use custom Displays to daisy-chain entity reference fields in this fashion (unless I misunderstand your requirements). 

To do this...


Install the Entity View Mode module.
Then in configuration - system - entity view modes, create a new display for the entity in question and make sure it is attached to the recording type. Give your new display a self-explanatory name such as "embedded recording" to avoid confusion later.
Next, go to your recording entity's display mode settings, (eg 'manage display') and in the new "embedded recording" display make sure the Recording Artist entity reference field is visible.
Finally, go to your Composition entity's display mode settings (its 'manage display'), and for the Recording entity reference field, choose rendered entity as its format and pick "embedded recording" as its view mode (this option becomes available if you click on the little gear icon on the right hand side). 


(Of course this would be far easier if you could add new display modes from the manage display setting! One day perhaps...)