Preheat oven to 400 degrees Fahrenheit and line a baking sheet with parchment paper.
Mix together the grated cheese and fresh rosemary.
Sprinkle 1/4 cup cheese and herb mixture onto the countertop in a rough 5 by 9-inch rectangle.
Roll out the puff pastry sheet onto the cheese and herb mixture until it’s 1/4 of an inch thick. If you are using frozen puff pastry, wait until it is thawed, unfold it and roll it a little to press the cheese into the dough.
Sprinkle another 1/4 cup of the cheese and herb mixture on top of the puff pastry sheet. Roll out until it is 1/8 of an inch thick.
Sprinkle with a 1/2 teaspoon of coarse sea salt.
Fold the puff pastry sheet in half, turn the dough so that the seem is on your left, and sprinkle with another 1/4 cup of the cheese and herb mixture.
Roll out the puff pastry sheet until it is 1/8 of an inch thick and 10 inches long.
Sprinkle with another 1/2 teaspoons coarse sea salt.
Cut the puff pastry sheet into 1 inch thick strips of dough.
Transfer the strips to the lined baking sheet, spacing them 1 inch apart. Twist the ends in opposite directions to give the dough a spiral. Press the ends into the parchment paper if they begin to untwist.
Chill for 30 minutes in the refrigerator.
Bake for 10 minutes at 400 degrees Fahrenheit. Lower the heat to 375 degrees Fahrenheit and bake for 15 minutes, until the straws are a deep golden color, puffed and crispy, and feel dry to the touch.
Let the straws cool briefly on the baking sheet until they are firm enough to transfer to a cooling rack. Store uncovered at room temperature.