Use an electric beater to beat the butter, caster sugar, brown sugar and vanilla in a bowl until pale and creamy. Beat in egg. Sift flour and cocoa over butter mixture. Add chocolate bits. Stir to combine. Turn onto a lightly floured surface. Knead gently until smooth. Shape into a disc. Wrap and place in the freezer for 20 minutes to rest.
Preheat oven to 180C/160C fan forced. Line baking tray with baking paper. Roll out the dough between 2 sheets of baking paper until about 1cm thick. Use a 9cm round cutter to cut a disc from dough. Use fingers to shape into a 12cm disc. Top with a mint pattie, face-side down. Fold over dough to enclose pattie and press to seal. Place seam-side down on the tray. Repeat with remaining dough and mint patties, re-rolling scraps of dough, to make 6 cookies. Bake for 15 minutes or until just firm. Cool slightly.
Drizzle with melted chocolate and sprinkle with peppermint crisp.