Preheat the oven to 350 degrees F.
Cut the cookie dough into 8 pieces. In bowl of electric mixer, on medium speed thoroughly combine the dough pieces, peppermint extract, 1/3 cup crushed peppermint candies, and 1/2 cup of the powdered sugar.
Roll the dough into 1-inch diameter balls. Arrange, 2 inches apart, on parchment lined cookie sheets and bake until set, about 8 to 10 minutes. Remove from the oven and let cool slightly on cooling racks.
Add 1/2 cup of the remaining powdered sugar to a small bowl. While the cookies are still warm, roll them in the sugar. Sprinkle with crushed peppermint candies and serve.
As the cookies cool, they will flatten slightly.
Preheat the oven to 375 degrees F.
Cut the roll of cookie dough in half. Put the halves in separate large mixing bowls. In 1 bowl, thoroughly mix 1/4 cup of the flour and the peppermint extract into cookie dough. In the second bowl, add the remaining 1/4 cup flour and enough food coloring to make a red dough. Mix well with gloved hands until the color is well blended.
Divide the white and red dough into 24 pieces each, (for a total of 48 pieces). Roll each piece into a rope 8 inches long and about 1/4-inch in diameter.
To shape candy canes, arrange 1 rope of each color side by side on parchment lined cookie sheets. Gently twist together and shape into a candy cane. Repeat with the remaining pieces, spacing the cookies about 2 inches apart. Bake until edges are very lightly browned, about 10 minutes. Remove from the oven and let rest, about 5 minutes, before transferring the cookies to racks. Gently remove them from the cookie sheets with a long metal spatula and cool on cooling racks.