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
1 1/2 teaspoons vanilla extract
2 1/2 cups quick-cooking oats

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