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.
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.
Heat oven to 375°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.
Pipe filling onto 36 cookies. Top with remaining cookies. Decorate with crushed candy canes as desired.