1. Whisk together flour, cocoa, baking soda, and salt. In another bowl, beat butter with both sugars on medium speed until fluffy, about 2 minutes. Add egg and peppermint extract; beat until well combined. Add flour mixture; beat just until combined. Mix in bittersweet chocolate. Transfer dough to plastic wrap, form into a disk, wrap tightly, and refrigerate until firm, at least 2 hours and up to 3 days; or freeze up to 1 month.

2. Preheat oven to 350 degrees. Scoop scant tablespoons of dough, roll into balls, and place 1 inch apart on parchment-lined baking sheets. Bake, rotating rack positions halfway through, until cookies are puffed and just set at edges, 10 to 12 minutes. Let cool on sheets 5 minutes, then transfer to a wire rack and let cool completely.

3. Reduce oven temperature to 300 degrees. Place starlight mints 1 inch apart on another parchment-lined baking sheet. Bake until soft to the touch and just beginning to melt, 8 to 10 minutes. Remove from oven; immediately top with another sheet of parchment that's been lightly sprayed with cooking spray. Press firmly with another baking sheet or a large, heavy skillet to flatten mints into thin disks slightly smaller than cookies. Let cool completely on tray. Remove parchment. (Mints can be stored in an airtight container at room temperature until ready to use.)

4. Spoon 1/4 teaspoon white chocolate onto center of each cookie. Top with a flattened mint, pressing slightly to adhere. Let stand until chocolate sets, about 1 hour; or refrigerate 5 minutes. Store in an airtight container at room temperature up to 1 week.