Preheat oven to 350°F.
Beat together butter, peanut butter, and sugar with an electric mixer until pale and fluffy, 2 to 3 minutes. Beat in egg, then remaining ingredients until a dough forms.
Arrange level tablespoons of dough 1 1/2 inches apart on 2 buttered baking sheets, then flatten in a crosshatch pattern to a 2 1/2-inch round with fork tines.
Bake in middle of oven until slightly puffed, 12 to 15 minutes. Cool on sheets 1 minute, then transfer cookies to a rack.
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