In a mixing bowl, cream together the butter and confectioners' sugar until light and fluffy, then add peppermint extract. Gradually beat in the flour and cornstarch.
Wrap dough in plastic wrap and chill for 30 minutes or so, until the dough is firm enough to work with.
Preheat oven at 350-degrees. Using a teaspoon scoop, dole out dough onto a cookie sheet. Use a spatula or the bottom of a glass to lightly compress the cookies, as they may not spread much while baking.
Bake for 9 to 11 minutes, until set. Let cool on sheet for 10 minutes, then move to a rack to finish cooling.
When the cookies are room temperature, prepare the frosting. Mash the two tablespoons butter until creamy; add the milk and peppermint extract. Beat in the confectioners' sugar until it is smooth. Add more sugar or milk, if necessary, to reach a thick yet spreadable consistency.
Use the back of a spoon to frost the cookies then sprinkle with peppermint candies. Store in an airtight container with waxed paper or parchment between layers. Makes about 25 teaspoon-sized cookies.
OM NOM NOM!