In a medium bowl, whisk together 1 cup all-purpose flour, 3/4 teaspoon salt, and 1/4–1/2 teaspoon cayenne pepper. (I used a rounded 1/4 teaspoonful, and that had just enough "kick" for me.) Add 8 ounces (2 cups) finely grated cheddar cheese and 1/2 cup (1 stick) softened unsalted butter; work with hands until a dough forms.
Transfer to a sheet of waxed paper. Form into 1 1/2-inch-diameter log; roll up tightly in paper. Refrigerate until firm, about 2 hours (or cover wrapped log in plastic and refrigerate up to 3 days or freeze up to 1 month—no need to thaw before continuing).
Preheat oven to 400°. Slice dough into 1/4-inch-thick rounds (discard ends if desired); arrange 1 inch apart on baking sheets. Bake until edges are golden, 16–18 minutes; transfer crisps to a rack to cool. (I stored them in an airtight container between layers of paper towels to absorb excess grease.)