Preheat oven to 350°F and line a baking sheet with parchment paper.
In the bowl of an electric mixer, cream together the butter, peanut butter, and sugar until light and fluffy. Beat in the egg. Add flour, baking powder, and salt and beat at low speed until just combined.
Drop by the tablespoon onto the baking sheet, then flatten with a fork, making a crosshatch pattern. Bake in batches until slightly puffed, 12 to 15 minutes. Cool cookies on a wire rack.