Preheat the oven to 350 degrees F (175 degrees C). Lightly grease baking sheets, or line with parchment paper.

Beat butter and sugar together in a large bowl with an electric mixer until light and fluffy. Beat in egg until completely incorporated. Mix in flour and salt to form a soft dough. Fold in crushed candy canes. Roll dough into walnut-size balls and place 1 inch apart onto prepared baking sheets.

Bake in the preheated oven until edges are golden, 8 to 10 minutes. Cool on the baking sheet briefly before removing to a wire rack to cool completely, 8 to 10 minutes more.

Meanwhile, make the icing: Whisk confectioners' sugar and warm water together in a small bowl until smooth. Dip the top of a cookie into icing, then sprinkle with crushed candy canes. Repeat with remaining cookies. Allow icing to set before serving, at least 5 minutes.