Threads, tasks, and the Objective-C SDK

What I need to do is copy an entire directory tree, one file at a time.  The easiest way to do that is to write a method that handles one folder, and then make that method call itself recursively when a subfolder is found.  But such code is not suitable for the UI thread, which must be event-driven and non-blocking.  If I were to do it using only UI code, I would have to implement a state machine, and recursion would be much harder.  But if I could call a blocking (but interruptable) API function for each file access, I could do the whole thing in one procedural thread.