Quest: The Finest Down
ID: 9420

After completing this quest, there should be a cut scene where he "gives" you the hat and he appears with a hatless model.

Mirren Longbeard: You treat my hat well now, <name>.  You won't find a finer one anywhere in this world or another.


https://youtu.be/sDZBIaaKSgg?t=198
https://youtu.be/0P7l8EiA8TA?t=215