Preheat oven to 350F/180C. Line a baking sheet with parchment. Set aside.
In a large bowl, beat butter, brown sugar, peanut butter and butter until creamy. Beat in the eggs and vanilla.
Mix flour, baking soda, baking powder, coconut and oats in another bowl; mix well. Add this to the peanut butter mixture, beating at low speed until well blended. Chill for 30 minutes.
Drop by level tablespoon onto prepared sheets spacing 2-inches/5 cm apart. Bake 10-12 minutes or until edges are beginning to get golden brown. Cool on sheets 2 minutes then remove to wire rack to cool completely.
Beat all filling ingredients together until the mixture is creamy. You may have to add a drop or 2 more milk to get the proper consistency--it should be like a soft frosting.
Place about 1-2 tablespoons of filling on half the cookies (on the side that's flat that was the down side on the sheet) then top with another cookie and press so the filling shows around the edges.