Heat oven to 350°F and line a 9-by-13-inch pan with parchment.
Beat butter, 1/4 cup peanut butter and both sugars, then beat in egg and vanilla.
Stir in flour, oats, baking soda and salt and press the thick dough evenly into the pan.
Bake for 16 to 18 minutes until the edges brown lightly, then cool for 15 minutes.
Spread the remaining peanut butter in a thin layer over the warm base and cool completely.
Whisk milk, confectioners' sugar and cocoa into a soft frosting, spread it over the bars and cut into 24 squares.