Preheat oven to 425. In a food processor fitted with the grating blade, grate cheese. Switch to cutting blade; add flour, butter, mustard powder, salt, ground pepper, and cayenne. Pulse until a dough forms.
Transfer dough to a lightly floured work surface; divide into 4 equal pieces. Roll each piece into a rope, about 3/8 inch in diameter and 18 inches long; cut each rope into six 3-inch pieces. Place pieces 1 inch apart on a baking sheet. Drag the tines of a fork down each straw to create ridges.
Bake until golden and firm to the touch, 15 to 20 minutes. Transfer to a wire rack to cool completely.
Store straws in an airtight container up to 2 days.