1. In a large bowl, stir together the sugar, softened butter, milk, vanilla, peppermint extract, and egg until well combined.
2. Gradually add the flour, baking powder, and salt, mixing until a dough forms.
3. Divide the dough in half; stir red food coloring into one half until evenly mixed.
4. Cover both dough portions and refrigerate for at least 4 hours to firm up for easier shaping.
5. Preheat oven to 375°F (190°C).
6. Combine crushed peppermint candies and sugar in a small bowl; set aside.
7. On a floured surface, roll a rounded teaspoon of white dough and red dough each into 4 to 5-inch ropes. Place the ropes side by side, press lightly together, and gently twist to form the candy cane stripes. Curve the top to form the traditional cane handle shape.
8. Place cookies on an ungreased baking sheet about 1 inch apart. Bake 9 to 12 minutes or until set and lightly browned.
9. Immediately sprinkle the warm cookies with the peppermint-sugar mixture so it sticks well.
10. Transfer cookies to a wire rack and cool completely, about 30 minutes.