Process the cheese, flour, cornstarch, salt, cayenne, and paprika in a food processor until the mixture resembles coarse meal, about 30 seconds.

Add the chilled butter and pulse until the mixture looks like damp sand.

Add the water and process until the dough gathers into a ball, about 10 seconds.

Transfer the dough to a work surface and divide it in half. Roll each half into a log roughly 10 inches long and 1.5 inches thick.

Wrap the logs tightly in plastic wrap and refrigerate until very firm, at least 1 hour.

Heat the oven to 350°F with a rack in the middle position and line two baking sheets with parchment paper.

Slice the chilled logs into 1/4-inch thick rounds using a sharp knife.

Arrange the coins on the prepared baking sheets, spaced about 1/2 inch apart.

Bake until the edges are lightly golden and the crackers are crisp, 15 to 18 minutes, rotating the sheets halfway through for even browning.

Cool the coins completely on the baking sheets to ensure they crisp up fully before serving.