For the vanilla-mascarpone whipped cream: Place mascarpone, 2/3 cup sugar, and vanilla into a mixing bowl and beat together. Set aside. Pour cream and remaining sugar into another mixing bowl until stiff peaks just form. Fold the slightly sweetened whipped cream into the mascarpone mixture until just combined.
Spread a very thin layer of whipped cream onto the surface of a 9-inch springform pan and top with a layer of cookies, breaking up a couple of cookies and fitting the pieces into the larger gaps.
Top the cookies with another layer of whipped cream and repeat with cookies and more whipped cream until all the cookies have been used.
Refrigerate icebox cake for at least 12 hours, or overnight.
Remove from the refrigerator, unmold from the springform pan, and frost with remaining whipped cream.
Top the cake with the extra chocolate chip cookie crumbs and grated chocolate and serve.