Process Cheddar cheese, flour, butter, 1/4 cup Parmesan cheese, cayenne pepper, and salt in a food processor until large crumbs form; transfer to a bowl. Stir milk into the cheese mixture until it forms a ball of dough.
Roll dough out onto a lightly floured work surface to about 1/8-inch thick. Use a pizza cutter or thin knife to trim the edges of the dough to leave a large square with sides measuring 7 to 8 inches. Cut dough into 1/3-inch strips and arrange onto a baking sheet.
Bake in preheated oven until puffy, about 15 minutes. Sprinkle 2 tablespoons Parmesan cheese over the warm cheese straws.
