In a large bowl, beat cream until soft peaks form. Add sugar and vanilla; beat until stiff. Spread heaping teaspoonfuls on the cookies. Make six stacks of cookies; turn stacks on edge and place on a serving platter, forming a 14-in.-long cake.
Frost top and sides with remaining whipped cream. Garnish with chocolate curls if desired. Refrigerate for 4-6 hours before serving.