Mix butter and sugar together until sugar is well incorporated. Beat in egg, then flour and cornstarch until dough comes together. Mix in cacao nibs.
Turn dough out onto a sheet of plastic wrap and form into a long log, about 2 inches in diameter.
Wrap in plastic and put in the refrigerator to firm up for at least 30 minutes.
Preheat oven to 350.
Remove dough from fridge and slice into cookies approximately 1/4-inch thick. Arrange on ungreased cookie sheets.
Bake for 10-12 minutes until just starting to get firm but not browned. Let cool on cookie sheets before removing to racks.
Once cookies are cooled, melt chocolate slowly either over a double boiler or in the microwave. If using the microwave, be sure to check after every 30 seconds or so and stop once chips are almost completely melted. Stir until chocolate is smooth, then add in peppermint oil and stir to smooth again.
Dip half the cookie into the chocolate, gently shake off excess, and set on wax paper to cool. Alternately, you can fill a pastry bag fitted with a round tip with the chocolate and pipe designs on the tops of the cookies.