In a food processor, grate sharp cheddar and chilled stick of butter- doing this in batches if necessary. Put cheddar mixture in a large bowl, covered with plastic wrap and leave overnight to soften- do not refrigerate at this stage. Sift together flour, cayenne, paprika and salt (if using unsalted butter). With clean hands, mix dry ingredients into softened cheese and butter- mixing very well until mixture is smooth. On ungreased sheet pan, in batches, put dough through a cookie press with a star tip in approximately 4 inch strips. (If you don't have a cookie press- the dough can be rolled with a bit of extra flour and cut into narrow strips.) Placement is approximately 1/2 inches apart. When sheet pan is filled, chill the pressed dough briefly to retain better shape as they bake.
Preheat oven to 350. Bake 15 minutes, checking after 12 minutes. Cheese Straws should be dried out but not browned. Remove to a wire rack to cook. Keep in an airtight container. Makes 4-6 dozen.