Beat the butter and white sugar with an electric mixer in a large bowl until smooth. Beat egg into butter mixture until completely incorporated. Mix flour and salt into the butter mixture until just incorporated. Fold crushed candy canes into the batter, mixing just enough to evenly combine. Roll dough into balls 1 tablespoon at a time; arrange on baking sheets.
Bake in the preheated oven until firm, 8 to 10 minutes. Allow cookies to cool on the baking sheet for 1 minute before removing to a wire rack to cool completely.
Whisk confectioners' sugar and warm water together in a small bowl until you have a smooth icing. Dip top of each cookie in the icing, Top with additional crushed candy cane, if desired. Set aside to let the icing dry, at least 5 minutes.