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Preheat oven to 350°. Line baking sheets with silicone liners or parchment paper.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt. Set aside.
Using a handheld mixer, beat butter, sugar, and brown sugar until smooth. Add peanut butter and mix just until blended. Add egg and vanilla, and mix just until blended. Do not overmix.
Gradually add flour mixture. Mix just until blended.
Drop dough by tablespoonfuls onto prepared pans. Place a chocolate wafer in the center of each cookie and gently press down into the cookie. Bake 14-16 minutes, or until lightly browned.
Cool cookies on pan for about 5 minutes. Then, transfer to wire racks to cool completely.
Makes about 40 cookies.