Grease and flour a 9 x 13 inch baking dish. (Or spray with baking spray.) Set it aside for now.
In your mixing bowl, beat together the butter and sugar until they are pale and fluffy.
Add in the egg, vanilla, and peppermint extract. Beat again for just a minute.
Add in half of the flour and mix in well.
Mix in the milk.
Add in the rest of the flour and mix well. Be sure to scrape the bowl with a rubber spatula to incorporate all of the ingredients.
Stir in the crushed candy canes. Your dough will be very stiff.
Spread the mixture evenly into your baking dish.
HINT: Press the dough down with clean, damp hands. This will allow you to handle the dough without it sticking to you. Alternatively, placing your hand inside a plastic sandwich bag also works.
Bake for 20-25 minutes, or until the cookie is done the middle and very lightly golden around the edges.
Cool completely.
Sprinkle the top with a dusting of powdered sugar and cut into bars.