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.
Variation: Peppermint Candy Cane Twists
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.