Grate the cheese on the medium holes of a grater to make 4 cups. Cover the grated cheese and leave out with the butter until at room temperature, at least 2 hours and up to overnight. (Note: You need to grate your own cheese here--no pre-grated.)
When ready to make the cheese straws, preheat the oven to 350 degrees F. Line 2 baking sheets with silicone baking mats or parchment paper.
Mix the flour, salt, cayenne, mustard and black pepper in a medium bowl.
Put the cheese and butter in the bowl of a stand mixer and beat with the paddle attachment until fairly smooth. Slowly beat in the flour mixture. Check the consistency; it should be just stiff enough to pipe. If not, add a little more flour.
Scrape the dough into a piping bag or resealable plastic bag. Use a 1/4-inch tip or cut a 1/4-inch hole in the plastic bag. Pipe the dough onto the prepared baking sheets in 5- to 6-inch strips. Bake until golden brown, 15 to 17 minutes. Let cool on a rack.