Preheat oven to 350°F. Line a 12-cup muffin pan with paper liners.
Whisk flour, sugar, cocoa, baking powder, baking soda, and salt in a large bowl.
Beat eggs, milk, oil, vanilla, and peppermint extract in a separate bowl until smooth.
Pour wet ingredients into the dry mix. Stir just until no dry streaks remain.
Stream in the hot coffee and whisk gently until the batter is thin and glossy.
Divide batter among liners, filling each about two-thirds full.
Bake 17–19 minutes, until a toothpick comes out with a few moist crumbs.
Cool cupcakes in the pan 5 minutes, then transfer to a rack to cool completely.
Beat softened butter on medium-high until creamy, about 2 minutes.
Gradually mix in powdered sugar and salt on low. Beat in peppermint and vanilla.
Add cream a little at a time, whipping until light and fluffy, 1–2 minutes more.
Swirl frosting onto cooled cupcakes. Sprinkle with crushed candy canes and sprinkles.