1 cup unsalted butter, at room temperature
1 1/3 cups sugar
3 cups flour, plus more for rolling
1/2 teaspoon baking powder
1/2 teaspoon salt
1 egg plus 2 egg yolks
2 teaspoons vanilla extract
1/2 teaspoon peppermint extract, divided
20 to 25 drops red food coloring
15 to 20 drops green food coloring
10 ounce white chocolate, finely chopped
5 ounces peppermint candy canes or peppermint candies, finely crushed
1. In the bowl of a standing mixer fitted with a paddle attachment, beat butter and sugar until pale and fluffy, about 4 minutes. Meanwhile, sift flour, baking powder, and salt into a separate bowl.
2. With mixer running, add egg and yolks one at a time, beating well after each addition. Add vanilla. Slowly add flour mixture and beat on low just until combined.
3. Divide dough into thirds. Shape one third into a disk; set aside. Return another third to mixer, raise speed to medium-low, and add 1/4 tsp. peppermint extract and red food coloring; shape into a disk and set aside. Clean mixing bowl and paddle attachment. Put last third of dough in mixing bowl and beat in remaining 1/4 tsp. peppermint extract and green food coloring; shape into a disk and set aside.
4. Draw a 6- by 8-in. rectangle on a sheet of waxed paper. Turn sheet over and lightly flour it. Working with one disk at a time, put dough in center of rectangle. Using a lightly floured rolling pin, your fingers, and a pastry scraper or a ruler, roll and shape each disk to fit rectangle. Layer each dough rectangle between sheets of waxed paper and chill at least 30 minutes and up to overnight. Let sit at room temperature for 5 minutes before continuing.
5. Peel waxed paper from rectangles and put one sheet on counter; discard the rest. Set green dough on waxed paper. Top with red dough, lining them up as evenly as possible, and gently press down. Top red dough with plain dough. Gently roll dough stack with a rolling pin to seal layers, then trim uneven edges with a very sharp knife.
6. Cut dough stack lengthwise into three 2-in.-wide columns. Cut each column crosswise into 1/3-in.-thick pieces. Lay pieces 2 in. apart on 3 parchment-lined baking sheets. Freeze 30 to 45 minutes. Meanwhile, preheat oven to 350°.
7. Bake cookies 7 minutes, then switch positions of baking sheets and bake until bottoms are light golden brown, 7 minutes more. (You may need to bake in 2 batches if your oven won't hold 3 baking sheets; keep cookies in freezer until ready to bake.) Cool cookies on wire racks.
8. Put chopped chocolate in a heatproof bowl that will fit over a pot with a few inches of water in it (bottom of bowl should not touch the water); set bowl aside. Bring water to a boil uncovered, turn off heat, and set bowl over water. Stir chocolate until melted and smooth. Dip each cookie about 1/2 in. into chocolate, then dip into or sprinkle with crushed candy. Set on waxed paper to harden (or chill in refrigerator 15 minutes).
Notes: Makes about 70 cookies. Store cookies, peppermint ends up, in an airtight container at cool room temperature or in refrigerator for up to 1 week. Freeze cookies for up to 2 months.