How should I deal with "thawed frozen puff pastry"?
No, you don't have to freeze it at all. 

The recipe you linked is indeed for puff pastry. It is rather tedious to create it, so it is available in supermarkets as a pre-fabricated food, just like pizza dough and other doughs. But it doesn't have a long shelf life in the fridge, so it is sold frozen. The recipe assumes that you will buy it frozen, and warns you to let it come to room temperature before you start baking. 

If you are making your own, the end product can be used in cookies immediately. Just pay attention to follow proper technique and work with a very cold dough and butter while making the puff pastry. The link you posted probably explains it - if not, search for better instructions.