Since this dough keeps well, you can slice and bake just what you need savory snacks to serve with wine or cocktails.
In a medium bowl, whisk together all-purpose flour (spooned and leveled), salt, and cayenne pepper. Add finely grated cheddar cheese and softened butter; work with hands until a dough forms.
Transfer to a sheet of waxed paper. Form into a 1 1/2-inch-diameter log; roll up tightly in paper. Refrigerate until firm, about 2 hours.
Preheat oven to 400 degrees. Slice dough into 1/4-inch-thick rounds (discard ends, if desired); arrange 1 inch apart on baking sheets. Bake until edges are golden, 16 to 18 minutes; transfer crisps to a rack to cool.
To store, cover wrapped log in plastic wrap; refrigerate up to 3 days, or freeze up to 1 month. Do not thaw before baking.