Beat butter, peanut butter, white sugar, and brown sugar with an electric mixer in a large bowl until smooth; beat in eggs.

Sift flour, baking soda, baking powder, and salt into a separate bowl; stir into butter mixture until dough is just combined. Chill cookie dough in the refrigerator for 1 hour to make it easier to work with.

Preheat the oven to 375 degrees F (190 degrees C).

Roll dough into 1-inch balls and place 2 inches apart onto ungreased baking sheets. Flatten each ball with a fork, making a crisscross pattern.

Bake in the preheated oven until edges are golden, about 7 to 10 minutes. Cool on the baking sheets briefly before removing to a wire rack to cool completely.