In the bowl of a stand mixer, cream the butter and sugar until light and fluffy. Add in the egg, peppermint extract, vanilla extract, and salt. Mix to combine.
Add the flour to the butter mixture and mix until combined.
Divide the dough in half. Wrap one piece in plastic wrap and place it in the refrigerator to chill. Divide the other piece into half again. Add green food coloring to one portion of the dough and red food coloring to the other portion. Wrap each dough separately in plastic wrap and then place them in the refrigerator to chill. The dough should be chilled for about 2 hours.
Preheat your oven to 350 degrees F. Divide the plain dough into 36 equal portions. Then, divide the red and green dough into 18 portions each. Roll each portion of all the doughs into a 4-inch rope.
Place a colored rope next to a plain rope, press together gently, then twist. Curve the dough at the top to create a candy cane shape. Repeat with the remaining ropes of dough. Place 2 inches apart onto an ungreased baking sheet.
Bake for 10-12 minutes or until the bottom edges of the cookie just start to turn golden brown and the cookie is set. Allow the cookies to cool on the baking sheet for about a minute before removing and placing them on a wire rack to cool completely.
Melt the chocolate chips in the microwave by heating in 30-second intervals. Stir between each interval. Continue heating and stirring until smooth. Dip the ends of the cooled cookies into the melted white chocolate. Place on a piece of wax paper and sprinkle with coordinating crushed peppermint candies.
Once the chocolate is set, store the cookies in an airtight container.