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Process cheese, flour, cornstarch, mustard, salt, and cayenne in a food processor until mixture is well combined and forms small beads, about 30 seconds. Add butter in 3 additions, pulsing 5 to 6 times after each addition, until mixture is the texture of wet sand, stopping to scrape down sides as needed. With processor running, pour 3 tablespoons water, 1 tablespoon at a time, through food chute, processing just until dough comes together in a mass, 20 to 25 seconds. (Be sure to stop processing once dough forms — you can continue to knead cheese mixture into a ball on a clean work surface if you’re unsure. It is important not to work it to death.)
Divide dough in half; roll each half into a 9- to 10-inch log on a clean work surface. Wrap logs tightly in plastic wrap. Transfer dough logs to refrigerator; chill until firm, at least 1 hour or up to 3 days.
Preheat oven to 375°F with racks in middle and lower third positions. Line 2 rimmed baking sheets with parchment paper.
Unwrap 1 dough log; slice crosswise into 1/4-inch-thick coins to yield about 30 coins, and arrange about 2 inches apart on prepared baking sheets.
Bake in preheated oven until centers of cheese coins are set but not crisp and edges and bottom sides of coins are golden brown and crisp, 14 to 16 minutes, rotating baking sheets between middle and bottom racks halfway through baking time. Remove cheese coins from oven, and transfer to a wire rack. Let cool completely, about 15 minutes. (Cheese coins will continue to crisp as they cool.)
Line baking sheets with fresh parchment paper, and repeat steps 4 and 5 with remaining dough log. Serve immediately.
12 ounces sharp cheddar cheese, shredded (about 3 cups)
1 1/2 cups all-purpose flour (about 6 3/8 ounces)
1 tablespoon cornstarch
1 1/4 teaspoons dry mustard
1 teaspoon kosher salt
1/4 teaspoon cayenne pepper
1/2 cup unsalted butter (4 ounces), chilled and cut into 1/2-inch pieces
3 tablespoons water