For the cookie dough
2 cups all-purpose flour
2 teaspoons baking powder
4 teaspoons baking soda
1 pound (4 sticks) butter, at room temperature
2/3 cup creamy peanut butter, preferably Skippy
1 cup granulated sugar
1 cup firmly packed light brown sugar
2 large eggs
1½ teaspoons vanilla extract
1/3 cup coarsely chopped peanuts
2½ cups quick-cooking oats

For the filling
¼ pound (1 stick) butter, at room temperature
½ cup creamy peanut butter, preferably Skippy
1 2/3 cups confectioners’ sugar