On a lightly floured work surface, roll out one sheet of puff pastry into a 14-by-10-inch rectangle. Cut lengthwise into twenty 1/2-inch-wide strips. Repeat process with remaining sheet of puff pastry.
Line a baking sheet with parchment paper or a nonstick baking mat. In a small bowl, whisk together egg with 1 tablespoon water. Place one hot dog half on the narrow end of one strip of puff pastry. Wrap hot dog in puff pastry, spiraling down to enclose like a mummy; brush with egg mixture to adhere. Transfer to prepared baking sheet. Repeat process with remaining hot dog halves and puff pastry. Brush tops of wrapped hot dogs with egg mixture; transfer to refrigerator to chill for 15 minutes.
Preheat oven to 400 degrees.
Transfer baking sheets to oven and bake until puffed and golden, 20 to 25 minutes. Let cool slightly before serving with mustard.