¾ cup granulated sugar
1¼ cup (1½ sticks) room-temperature butter, divided
1 large egg
1 large egg yolk
2 tsp. peppermint extract, divided
2 pinches of kosher salt, divided
1½ cups all-purpose flour
½ cup crushed candy cane pieces, plus more for decoration
1½ tsp. cornstarch
1 tsp. baking powder
½ tsp. baking soda
2 cups powdered sugar
3 Tbsp. milk
Green food coloring
Preheat the oven to 350 degrees. Line an 8-inch square cake pan.
In a large mixing bowl, cream the granulated sugar and ½ cup butter with an electric mixer on high speed until fluffy and light, about 5 minutes.
Add the egg and yolk and beat on high speed, stopping once to scrape down the sides and bottom of the bowl. Add 1 teaspoon peppermint extract and a pinch of salt and beat until combined.
Add the flour, candy cane pieces, cornstarch, baking powder and baking soda and fold in with a spatula until even and there are no white streaks.
Pour the batter evenly into the cake pan. Bake 25 minutes, until it is browned on top but still soft. Let cool completely.
To make the frosting, place the powdered sugar, the remaining ¾ cup butter, milk, the remaining 1 teaspoon peppermint extract and a few drops green food coloring in a bowl. Beat the ingredients together on medium-high speed until light and fluffy. Add more peppermint extract or food coloring to taste.
Spread the frosting in a thick even layer atop the cookie and sprinkle with crushed candy canes.
Use a sharp knife to cut the cookie into 2-by-2-inch bars. Store in an airtight container at room temperature up to 5 days.