Line three 9-inch cake pans with foil, leaving about a 2-inch overhang on all sides. (See Cook's Note.)
Add the cookies and sugar to the bowl of a food processor and pulse to fine crumbs. Drizzle in the butter and pulse until the crumbs are completely coated and moistened with the butter. Firmly press the crumb mixture into the bottom of one of the prepared pans; set aside.
Add the dark chocolate pudding mix to a large bowl and whisk in 2 cups of the heavy cream. Continue whisking until light and fluffy, about 2 minutes. (It's fine if there are still a few granules in the mix; they will dissolve once set.) Pour the pudding mixture on top of the crumb crust and spread until even. Freeze until firm, about 2 hours.
Meanwhile, clean out the bowl, add the chocolate pudding mix and whisk in 2 cups heavy cream. Continue whisking until light and fluffy, about 2 minutes. Pour into another prepared pan and freeze until firm, about 2 hours.
Clean out the bowl again, then add the white chocolate pudding mix and whisk in the remaining 2 cups heavy cream. Continue whisking until light and fluffy, about 2 minutes. Pour into the third prepared pan and freeze until firm, about 2 hours.
When all 3 layers are completely frozen, use the foil to lift out the cakes from the pans. Place the dark chocolate layer crumb-side down on a cake stand or large plate. Top with the chocolate layer and then with the white chocolate layer. Refrigerate until the layers are thawed and the cake is set, about 8 hours and up to overnight.
Before serving, run a hot knife around the outside of the cake to smooth it and sprinkle the top with the chocolate shavings.
If you do not have three 9-inch cake pans, you can prepare the layers one at a time. Make and freeze the first cookie and pudding layer according to the recipe. When the layer is frozen, remove it from the pan, place it on a plate and return it to the freezer. Reline the cake pan and repeat with the chocolate and white chocolate layers.
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