1. Finely grind peppermint candies in a food processor.

2. Whisk together flour, cocoa, and 1/8 teaspoon salt in a bowl.

3. Beat butter and sugar with a mixer on medium speed until combined, about 3 minutes. Beat in egg, then vanilla, and beat for 1 minute. Reduce speed to low, and gradually beat in flour mixture until combined, about 2 minutes. Shape dough into a disk, and wrap in plastic wrap. Refrigerate until firm, at least 2 hours.

4. Roll out dough to a 1/8-inch thickness on a lightly floured surface. Transfer to a baking sheet, and refrigerate until firm, at least 30 minutes.

5. Preheat oven to 325 degrees. Cut out star shapes using 2 1/2-, 4 1/2-, and 5-inch cookie cutters. Cut a smaller star from the center of each cookie using cookie cutters that are about 1 inch smaller than the larger cutter used. Transfer cookies to parchment-lined baking sheets, spacing each 1 inch apart. Bake until slightly firm, 12 to 15 minutes.

6. Fill the center of each cookie with ground peppermints. Bake until candy melts, about 6 minutes. Let cookies cool on sheets set on wire racks for 10 minutes. Transfer cookies to racks, and let cool completely.