1Make the cookie crust:
In the bowl of a food processor, finely grind the cookies. Add the melted butter and process until the crumbs start to stick together. Press the crumb mixture firmly into the bottom and up the sides of a 9-inch-diameter springform pan with 2 3/4-inch-high sides. Chill until ready to use.
2Make the filling:
Position a rack in the center of the oven and preheat to 350°F.
3In a dry metal bowl set over a pan of barely simmering water, melt the chocolate, stirring until smooth. Let cool until lukewarm.
4In a stand mixer fitted with the paddle attachment, beat the cream cheese and sugar until blended. Add the eggs, one at a time, beating until just incorporated. Add the melted chocolate, along with the heavy cream, cocoa powder, and the vanilla and peppermint extracts and beat, scraping down the bowl and paddle as necessary, until smooth and streak-free.
5Pour the filling into the prepared crust, smooth the top, and bake until the center is just set, about 1 hour. Transfer to a rack to let cool for 20 minutes. Press down gently on the puffed edges to flatten then run a small sharp knife around the top edge of the pan to loosen the cake. Do not remove the pan sides.
6Let cool completely then cover with a double layer of plastic wrap and chill overnight. Bring to room temperature before serving, if desired.
The cheesecake can be baked ahead, cooled, wrapped in a double layer of plastic wrap, and refrigerated for up to 3 days.