Cream 10 tablespoons of the butter with the sugars in a medium bowl, and then beat in the vanilla and egg yolk. Add the flour and cocoa and beat until smooth. Transfer the dough to wax paper and roll into a 10-by-2-inch log. Wrap and refrigerate 1 hour.
Slice the dough into 1/4-inch rounds and flatten gently on a baking sheet. Bake 7 to 8 minutes, and then let cool 10 minutes.
Meanwhile, cream the remaining 4 tablespoons butter with the confectioners' sugar, peppermint extract and 1 teaspoon of the cream until smooth. Spread about 2 teaspoons of peppermint cream each over half of the cookies and close with a plain baked cookie.
Melt the chocolate chips with the remaining cream in a small saucepan, adding more cream to make it smooth but not runny. Spread over half of the cookies and top with the candy canes while the chocolate is still wet.