Calling Aura.Action attribute from Lightning Controller/Helper

I have not found this documented anywhere (just like you said in your original post), but I have found that you can accomplish this by using $A.enqueueAction(clickme). This also seems to fire immediately because there is no server connection involved so the framework must see that it does not actually need to go in the server queue.

As of API v41.0, Salesforce discourages calling Aura.Action directly and recommends using Events as the standard way to communicate between components.