Roll out puff pastry dough into a 10 x 12-inch rectangle. Dough should be ⅛-inch thick.
Make egg wash by whisking together egg, 1 tablespoon of water, and a pinch of salt in a small bowl.
Use a pastry brush to brush the egg wash in a thin layer all over the rolled out dough.
Sprinkle half the dough with half of the shredded cheese, then fold the uncovered side over the cheese-covered side.
Roll out again to 10 x 12.
Brush dough again with egg wash, and sprinkle with the remaining shredded cheese.
Then roll your rolling pin gently over the top, pressing in the cheese.
Line up the dough so that the long edge is on the top and bottom (horizontal). To make long breadsticks, cut dough vertically into slices that are close to 1-inch wide. To make short breadsticks, cut the dough in half horizontally (so you will have two 5 x 12-inch rectangles) before slicing into strips.
Pick up each strip, hold each end, and twist in opposite directions several times.
Place twists on baking sheet, leaving an inch of space between them. (You may need to bake these in two batches if they don’t all fit on your baking sheet.)
Paint any exposed pastry (not covered in cheese) with egg wash.
Bake for 15-20 minutes, until puffed and golden.
Notes
Working with Puff Pastry: Keep your puff pastry dough as cold as possible while you are working with it. If it starts to warm up, and is difficult to roll out or to twist into cheese straws, pop it back in the refrigerator for a few minutes before trying again. Freezing Instructions: To freeze cheese straws before baking (recommended), follow the instructions up through making the twists, then lay each twist on a parchment-lined small baking sheet. (They don’t need to be 1-inch apart, but they shouldn’t be touching.) Move the baking sheet into the freezer and freeze until twists are completely firm. Then, you can move the twists into a freezer-safe ziplock bag and keep frozen for up to 3 months. Bake straight from the freezer—you may need to add a couple of minutes to the baking time. You can also freeze baked cheese straws. Reheat by taking directly out of the freezer and baking in a 375-degree oven until warm (no need to defrost first). Make-Ahead Instructions: Cheese straws are at their most crunchy and delicious right after baking, and up to 1-2 hours afterwards. After that, they lose a little crunch. They will keep well at room temperature for a few days (stored in an airtight container), and can be reheated in a 375-degree oven for 3-5 minutes to bring back some of their crunch!