This recipe is from Yvonne Ruperti of America's Test Kitchen.
1 1/2 sticks unsalted butter
2 cups confectioners' sugar, divided
1 large egg
1 1/2 cups all-purpose flour, plus extra for work surface
3/4 teaspoon salt
1 large egg white
4 peppermint candy canes, crushed
In the bowl of standing mixer, cream butter and sugar on medium-high speed until light and fluffy, about 3 minutes. Reduce speed to medium and add egg until combined. Reduce speed to low and add flour and salt. Mix until dough is just combined. Form dough into a 4x4-inch square, wrap in plastic, and chill until firm, about 1 hour.
Adjust oven racks to lower-middle and upper-middle positions, and heat to 350°. Line 2 baking pans with parchment paper. Flour work surface lightly and roll out dough to one-eighth-inch thickness. Using a star-shaped cookie cutter, cut out cookies and place on prepared pan, about 1 inch apart.
Bake until edges are light golden-brown, about 12 minutes, rotating pans halfway through baking. Transfer trays to wire racks and cool completely.
Make the icing: In the bowl of a standing mixer fitted with a whip attachment, combine confectioners' sugar and egg white on medium speed. Mix in water, 1 teaspoon at a time, until mixture is consistency of honey.
Using a small offset spatula, spread icing onto each cookie and top with crushed candy canes. Let icing set before serving, about 4 hours.