Place the dark chocolate in a heatproof bowl set over simmering water and stir until melted. Remove from heat and let cool slightly.
In a large bowl, whip 1 cup heavy cream with 1 tablespoon sugar and 1/3 teaspoon vanilla until stiff peaks form. Fold the whipped cream into the melted dark chocolate to create the dark mousse.
Repeat the melting and whipping process with the milk chocolate using another 1 cup cream, 1 tablespoon sugar, and 1/3 teaspoon vanilla.
Repeat once more with the white chocolate, the remaining 1 cup cream, 1 tablespoon sugar, and 1/3 teaspoon vanilla.
Place the bottom cake layer in a loaf pan or small square pan lined with plastic wrap. Spread the dark mousse evenly over it.
Add the middle cake layer and spread the milk chocolate mousse on top.
Top with the final cake layer and spread the white chocolate mousse over the surface.
Cover and refrigerate at least 4 hours until the mousses are firm.
Lift the cake out using the plastic wrap, slice, and garnish with chocolate curls before serving.