In a large mixing bowl beat the butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, brown sugar, soda, and salt. Beat until combined. Add egg and vanilla. Beat until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour and the cocoa powder. Stir in peanut butter and milk chocolate pieces. Cover and chill dough for 2 to 3 hours or until easy to handle.

Preheat oven to 375 degree F. Roll dough into 1-inch balls. Place 2 inches apart on lightly greased cookie sheets. Flatten cookies slightly with a glass dipped in sugar. Bake for 7 to 8 minutes or until tops are cracked and look dry. Let cool on cookie sheets for 1 minute. Remove and cool completely on wire racks.

In a small bowl stir together frosting and peanut butter. Spread a scant 2 teaspoons frosting mixture on the bottoms of half of the cookies. Top with remaining cookies, bottom sides together. Store sandwich cookies in an airtight container at room temperature up to 24 hours or freeze in freezer container up to 3 months. Makes 24 sandwich cookies.