In a large bowl or using a stand mixer, beat 1 cup sugar, ½ cup butter, 1 tsp. vanilla, 1 egg, and 2 oz. of melted and cooled unsweetened chocolate until well combined.
Stir in 1 cup of flour and ½ tsp. salt until a dough forms.
Scoop large balls of cookie dough onto an ungreased cookie sheet, spacing them 2 inches apart.
Dampen the bottom of a glass, dip it in 2 Tbsp. sugar, and press down on a cookie dough ball until it’s ¼-inch thick. Repeat for each cookie, re-dipping the glass in sugar before flattening the next ball.
Bake the cookies at 375°F for 8 minutes, then transfer them to a cooling rack to cool completely.
While the cookies bake, prepare the peppermint frosting: Using a stand mixer, beat 2 ½ cups powdered sugar, ¼ cup butter, 3 Tbsp. milk, and ½ tsp. peppermint extract until smooth. If the frosting is too thick, add additional milk 1 tsp. at a time until it reaches piping consistency.
Crush candy canes in a food processor or with a rolling pin until finely crumbled.
Once the cookies are cool, transfer the frosting to a quart-sized storage bag. Seal the bag, remove any air, and snip a small corner off the bag. Pipe a swirl of frosting onto each cookie.
Prepare the chocolate drizzle: In a microwave-safe bowl, combine ¼ cup butter, 2 Tbsp. corn syrup, and 1 cup semisweet chocolate chips. Heat in 10-15 second intervals, stirring between each, until smooth. Alternatively, melt the mixture in a double boiler. Allow the chocolate to cool slightly before using.
Transfer the melted chocolate to a second quart-sized storage bag. Snip off a small corner and drizzle the chocolate over the frosted cookies.
While the chocolate drizzle is still wet, sprinkle crushed candy canes on top of each cookie so they stick.