Preheat oven to 350°F. Line a 12-count muffin pan with paper liners or grease well with non-stick cooking spray. Set aside.
In a medium bowl, combine flour, unsweetened cocoa power, baking soda, and salt. Set aside.
In a large bowl, whisk vegetable oil, milk, and egg. Add sugar, vinegar, red food coloring, vanilla bean paste (or extract), and peppermint extract. Whisk to combine.
Whisk flour mixture into wet ingredients until just combined. Spoon batter into prepared muffin tin, filling almost 3/4 full.
Bake for 20-24 minutes, or until toothpick inserted into center of cupcake is clean. Remove from oven and let cool completely before frosting.
To prepare frosting, cream softened butter in a large bowl until light and fluffy, for about 3 minutes. Add powdered sugar, 1/2 cup at a time, mixing to incorporate. Add milk and peppermint extract, beating to combine.
Pipe or spread over cooled cupcakes. Then top with crushed candy canes. Alternatively, you could stir candy into frosting and then pipe, if desired.