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Put the cold butter and Gouda, the cheddar, salt, black pepper and cayenne in a food processor and pulse until the butter is in bits and the mixture forms small curds. Add the flour and pulse again, in long pulses, until the dough is moist and forms larger, popcorn-like curds.
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Turn the dough out and knead it gently, just until you can shape it into a ball. Divide the dough in half and pat each half into a disk.
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Working with one disk at a time, place the dough between two pieces of parchment paper and roll to a thickness of 1/4 inch. Slide the dough onto a baking sheet - you can stack the slabs - and freeze for at least 1 hour.
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Center a rack in the oven and preheat it to 350 degrees F. Line a baking sheet with parchment paper or a silicone baking mat. Have a 1 1/2-inch-diameter cookie cutter at hand.
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Working with one piece of dough at a time, peel away the top and bottom papers and return the dough to one piece of paper. Cut out as many cookies as you can and place them on the lined baking sheet an inch apart. Gather the scraps, then combine them with the scraps you get from the second disk, re-roll, freeze, and bake.
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Bake the cookies for 16-18 minutes, roaring the baking sheet after 9 minutes, or until they’re lightly golden on top and more golden on bottom. Let the cookies rest on the baking sheet for a couple of minutes, then transfer them to a rack and cool completely.
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Repeat with the remaining dough, using a cool baking sheet.