Heat oven to 350°F. Line a 12-cup muffin tin with paper liners.
Set a whole cookie in the bottom of each liner for a crunchy base.
Whisk flour, sugar, cocoa, baking powder, baking soda, and salt in a large bowl.
Combine buttermilk, hot coffee, oil, vanilla, and vinegar in a measuring cup.
Pour wet mixture into dry ingredients and whisk until smooth and glossy.
Fold in the finely crushed cookies just until evenly distributed.
Divide batter among liners, filling each about two-thirds full.
Bake 18–20 minutes, until a tester comes out with a few moist crumbs.
Cool in the pan 10 minutes, then transfer cupcakes to a rack to cool completely.
Beat softened butter and cream cheese on medium-high until fluffy, 2–3 minutes.
Gradually mix in confectioners' sugar and salt, drizzling in cream until spreadable.
Fold in finely ground cookies, then beat 15 seconds to smooth. Pipe or spread onto cooled cupcakes.
Finish with extra cookie crumbs or a halved cookie on top.