1. In medium bowl, combine flour and cocoa. Set aside. In large bowl, with mixer on high, beat butter and sugar until fluffy, about 3 minutes. Beat in egg until blended. With wooden spoon, stir in flour mixture until dough forms. Divide dough in half, and shape each half into 9-inch log. Wrap and refrigerate 2 hours.Make filling:
2. In large bowl, with mixer on high, beat powdered sugar, butter, milk, and peppermint extract until fluffy. Place in resealable plastic bag, and snip off corner.
3. Heat oven to 375 degrees F. Slice cookie logs into 1/2-inch slices and place 1/2 inch apart on baking sheets. Bake 12 minutes. Cool 2 minutes and transfer to wire racks. Cool completely.
4. Pipe filling onto 36 cookies. Top with remaining cookies. Decorate with crushed candy canes as desired.