Preheat oven to 425°F. Sift together flour, mustard, salt, and cayenne in a medium bowl; set aside. Beat cheese and butter in a stand mixer fitted with the paddle attachment on medium speed until well blended and mixture becomes a spreadable consistency (this will help the flour mixture incorporate evenly), about 2 minutes. Reduce mixer speed to low, and gradually beat in flour mixture until completely incorporated. Add 2 tablespoons water, and beat on medium speed until dough comes together, about 1 minute.
Turn dough out onto a lightly floured work surface, and knead 5 times. Place dough on a large sheet of wax paper or parchment paper, and roll into a 12- x 9-inch rectangle (about ¼ inch thick). Slide dough and paper onto a baking sheet, and refrigerate until cold, about 15 minutes.
Line two baking sheets with parchment paper. Cut dough in half crosswise, and cut each half into 6-inch-long, ¼-inch-wide strips. Transfer dough strips to prepared baking sheets, spacing at least ½ inch apart. Using your fingertips, wiggle dough strips gently on baking sheets to create a slightly curvy shape.
Bake in preheated oven, one sheet at a time, until cheese straws are golden brown and crisp, 12 to 14 minutes. Let cool on baking sheet 10 minutes; transfer to a wire rack to let cool completely, about 15 minutes.
