Beat butter, peanut butter, applesauce, white sugar, and brown sugar with an electric mixer in a large bowl until smooth. Add eggs and beat until smooth.
Sift together flour, baking powder, and baking soda in a separate bowl; stir into peanut butter batter. Put batter in refrigerator for 1 hour.
Roll dough into walnut-sized balls and place 2 inches apart onto ungreased baking sheets. Flatten each ball with a fork, making a crisscross pattern.
Bake in preheated oven until cookies begin to brown, about 10 minutes.
