Crush the chocolate cookies into fine crumbs. Stir in the melted butter, press firmly into a 9-by-13-inch dish, and chill.
Beat the softened cream cheese, powdered sugar, and vanilla until smooth. Fold in half of the whipped topping and spread over the chilled crust.
Whisk the pudding mixes with the cold milk until thickened. Spoon over the cream layer and spread gently to the edges.
Spread the remaining whipped topping over the pudding. Dust with cocoa and add chocolate curls. Cover and refrigerate for at least 4 hours before slicing.