1. Preheat your oven to the appropriate temperature and line baking sheets with parchment paper.

2. In a mixing bowl, combine the melted butter, egg white, and peppermint extract. In a separate bowl, whisk together the flour, cocoa, and icing sugar. Gently mix the dry ingredients into the wet ingredients until a smooth dough forms.

3. Carefully roll the dough into small balls or logs, depending on your preferred shape. Arrange the shaped dough on the prepared baking sheets and bake according to the recipe instructions until the cookies are set and slightly crisp around the edges. Allow them to cool completely before proceeding to the next step.

4. In a mixing bowl, beat the softened butter, corn syrup, and a pinch of salt until smooth. Gradually add the icing sugar, peppermint extract, and a tablespoon or two of milk, mixing until the peppermint cream is creamy and spreadable.

5. Spread a generous amount of the peppermint cream over the cooled chocolate cookies, then carefully roll them up. If desired, insert the paper straws into the center of the rolled cookies for a fun and festive presentation.

6. Melt the dark chocolate and drizzle it over the assembled peppermint chocolate roll cookies. Sprinkle the crushed candy canes on top to add a beautiful finishing touch. Allow the chocolate to set before serving.