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In the bowl of an electric stand mixer, fitted with the paddle attachment, cream together the butter, cheese, salt, and pepper.
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With the mixer on low, slowly add in the flour. When the flour is incorporated, add the buffalo sauce. Mix on medium/low speed until the dough is a soft, sticky ball.
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Divide the dough in half. Press each half into a flat disc. Wrap each half in plastic wrap and refrigerate for 40 minutes to an hour.
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Preheat the oven to 375 degrees F.
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Roll each disc out on a well-floured surface until about 1/8 of an inch thick. Use a 1-inch cookie cutter or shot glass to cut circles out of the dough. Use a toothpick to pierce a hole in the center of each cracker. Place the crackers close together on large baking sheets lined with parchment paper. Bake for 12 – 15 minutes until golden around the edges.
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Crackers last 3 – 4 days at room temperature in an air-tight container.