Preheat oven to 350 degrees. In a medium bowl, whisk together flour and baking powder; set aside.
In a large bowl, beat peanut butter, sugar, and butter until smooth. Beat in egg. Gradually add flour mixture, beating to combine.
Pinch off dough by the tablespoon; roll into balls. Place 2 inches apart on parchment-lined baking sheets. Using a fork, press balls in a crisscross pattern, flattening to a 1/2-inch thickness.
Bake cookies, rotating halfway through, until lightly golden, 18 to 22 minutes. Cool cookies on a wire rack.
Store in an airtight container at room temperature up to 1 week.