Line 2 baking sheets with parchment paper; set aside.
Place a sheet of parchment paper on a flat surface and spray it lightly with nonstick spray. Lay the puff pastry on top and place a 2nd sheet of sprayed parchment over the puff pastry.
Use a rolling pin to slightly flatten the puff pastry into a 10x11-inch rectangle. Carefully remove the top parchment. Spread the pastry evenly with a thin layer of 2 tablespoons Dijon mustard. Sprinkle ½ cup shredded cheese over the mustard and top with 1/3 cup grated Parmesan. Lightly & evenly sprinkle the Parmesan with sweet paprika, kosher salt and ground black pepper. Replace the top parchment paper and press the cheeses & seasonings into the pastry by rolling the pin back and forth a few times.
Flip the parchment-encased pastry over and remove what is now the top parchment. In a small bowl, whisk together the egg and water. Use a pastry brush to evenly coat the top-facing side of pastry. Sprinkle with the remaining ½ cup shredded cheese and 1/3 cup grated Parmesan.
Replace the top parchment paper and, again, roll the pastry with the rolling pin to press the cheese into the pastry. Remove the top piece of parchment paper.
Use a large, sharp knife or a pizza cutter to trim off the rough edges of the dough. Cut the pastry into 14 equal (3/4-inch) strips the length of the dough. Beginning in the middle of each strip of dough, gently twist each strip of pastry and place them on the prepared baking sheet 1-inch apart. Refrigerate the dough for 30 minutes.
While the dough cools in the fridge, adjust the oven rack to the middle positions and preheat the oven to 425°F.
Place the cooled cheese straws on the baking sheet to the oven and bake 12-16 minutes or until golden brown and fully puffed, rotating the baking sheet halfway through baking. (The cheese straws will crisp a little after removing from the oven.) Cool 5 minutes on the baking sheet before serving.
Serve with honey mustard, cheese sauce or your favorite dipping sauces. Enjoy!