1. Preheat oven to 350°F (175°C) and line a 12-cup muffin pan with paper liners.

2. In a bowl whisk flour, cocoa powder, baking powder, baking soda, and salt until combined.

3. In another bowl cream 1/2 cup butter and sugar until light and fluffy.

4. Beat in eggs one at a time, then add vanilla and peppermint extracts.

5. Fold in sourdough starter and buttermilk until just combined.

6. Alternate adding dry ingredients and hot water or coffee to the wet mixture, beginning and ending with dry ingredients.

7. Divide batter among liners, filling each two-thirds full, and bake 18–20 minutes until a toothpick comes out clean.

8. Cool cupcakes completely on a wire rack before frosting.

9. Beat 1 cup butter until creamy, then gradually add powdered sugar alternately with heavy cream until smooth.

10. Stir in peppermint and vanilla extracts, adjusting consistency with cream or sugar.

11. Fill a piping bag and swirl frosting on cooled cupcakes, then sprinkle with crushed candy canes.