Place the cream in a large mixing bowl and beat with an electric mixer until it is fluffy and soft. Add the sugar and vanilla. Continue beating until the cream holds stiff peaks. (When you lift a beater or whisk straight up, the whipped cream should form a peak that stands up.)
Spread a scant tablespoon of whipped cream on each cookie. Press cookies together to make 3-inch stacks. Spread an inch wide line of whipped cream down a plate or serving platter.
Assemble cookie stacks into a log on the plate along the line of whipped cream. Generously frost the cookie log with remaining whipped cream and sprinkle with grated chocolate.
Cover tightly with saran wrap and refrigerate overnight. To serve, slice diagonally to create striped pieces.
RECIPE NOTE: Alternatively, you can make mini cakes in small loaf pans, as shown in the above photo.