How to Make It
Sift the flour into a bowl with the cocoa, baking soda, baking powder and salt. In another bowl, using an electric mixer, beat the butter with the sugar until creamy. Add the melted chocolate, egg and vanilla and beat until smooth. Beat in the dry ingredients until incorporated. Pat the dough into 4 disks, wrap in plastic and chill until firm, at least 20 minutes.
Preheat the oven to 375° and line 4 baking sheets with parchment paper. On a sheet of lightly floured parchment, roll out one disk of dough 1/4 inch thick. Using a 2 1/2- to 3-inch cookie cutter, stamp out as many cookies as possible; transfer to a baking sheet. Gather the scraps and roll out more cookies; if the dough gets too soft, refrigerate briefly. Chill the cut cookies for 5 minutes. Repeat with the remaining 3 disks of dough.
Working in batches, bake the cookies in the middle and lower thirds of the oven for about 12 minutes, until puffed; shift the pans halfway through baking. Transfer to racks to cool.
In a bowl, using an electric mixer, beat the butter, marshmallow crème and confectioners’ sugar until fluffy. Beat in the peppermint extract.
Pipe the peppermint filling onto the underside of half of the cookies. Top with the remaining cookies, pressing to spread the filling to the edges. Refrigerate the cookies for 20 minutes, or until the filling is set.
Put the bittersweet and white chocolates in 2 separate microwave-safe bowls. Microwave at high power in 30-second bursts until melted. Stir 1/4 teaspoon of the oil into each.
Drizzle the melted chocolate over the cookies or dip them in the bittersweet chocolate or white chocolate (or both); return the cookies to a sheet of parchment paper and refrigerate until set, about 20 minutes. Serve.