Combine flour, salt, paprika, and cayenne pepper in a food processor fitted with the metal blade; pulse to combine. Add butter; pulse until mixture resembles coarse meal. Add cheese; process until dough starts to hold together.
Turn out dough onto a lightly floured surface; knead a few times. Divide into 4 equal pieces, and roll into logs, each about 6 inches long and 1 1/4 inches in diameter. Wrap in plastic; refrigerate until firm, at least 1 hour or up to 3 days. Dough can be stored in the freezer at this point for up to 1 month.
Preheat oven to 350 degrees. Line baking sheets with Silpat baking mats. Cut dough into 1/3-inch-thick slices; place 2 inches apart onto prepared baking sheets. Bake until lightly browned, about 20 minutes. Remove from oven; let cool on baking sheet 1 minute, then transfer to a wire rack to cool completely.