Heat oven to 350 degrees F. Line a large baking sheet with nonstick foil. Set aside.

In a large bowl, knead together cookie dough, 1/4 cup of the crushed candy canes, the peppermint extract and flour to incorporate all of the ingredients.

Break off a level teaspoon of dough and roll into a 4-1/2 to 5-inch-long rope. Transfer to prepared baking sheet, spacing each rope at least 1 inch from the others. Repeat until sheet is full; refrigerate dough in between batches.

Bake at 350 degrees F for 7 minutes, until puffed and just beginning to color. Transfer to a wire rack to cool.

Stir together confectioners' sugar and 4 teaspoons water until smooth to form a glaze. Brush glaze over wands and let dry. With a small paintbrush, use remaining glaze to paint stripes diagonally across wands, spacing about 1/2 inch apart. Sprinkle wet stripes with some of the remaining crushed candy to resemble a candy cane. Let dry completely. Store in airtight container up to 2 weeks.

If you cannot find peppermint sugar, you can crush candy canes and use those instead: You will need 1/4 cup crushed canes for the dough and 1/4 cup to make the stripes (8 canes total).