To prepare the cake, preheat oven to 350F. Grease an 8-inch cake pan and line bottom with parchment paper.
In a large bowl, mix together sugar, flour, cocoa powder, baking powder, baking soda and salt.
Crack in the egg, spoon in the sour cream, and pour in the oil and the vanilla. Mix with a wooden spoon until just combined.
Pour in the boiling water, and slowly mix until smooth and liquidy.
Pour batter into the prepared cake pan. Bake for 20 to 25 minutes or until a toothpick inserted into the cake comes out clean.
Place the cakes (in the pans) on a wire rack to cool. Then invert the cakes onto a greased rack. Cool completely. Then slice cake layers in half horizontally. Reserve one layer for future use.
To prepare ice cream, let ice cream sit in room temperature for 10 minutes to soften. Scoop 3 cups of vanilla ice cream, and mix in with the strawberry puree to make strawberry ice cream. Add mango puree in another 3 cups of vanilla ice cream to make mango ice cream.
To assemble, lightly grease a 3-quart bowl and line with plastic wrap with overhang long enough to cover the top. Fill the base of the bowl with strawberry ice cream, followed by mango ice cream and the remaining vanilla ice cream. (if ice cream becomes too soft, chill each layer in the freezer for 10 minutes before adding another layer)
Top the ice cream with the chocolate layer cake. Cover the cake with plastic wrap overhang. Return it to the freezer and freeze at least 4 hours.
To prepare frosting, in a large heatproof bowl, beat sugar with a handheld electric mixer for a minute until foamy. Increase speed and add sugar slowly while beating. Continue beating until meringue becomes stiff and glossy.
Remove the ice cream cake from the freezer. Invert the cake onto a serving plate and wrap a hot kitchen towel around the bowl for 20 seconds. Loosen the bowl and remove the plastic wrap.
Smear the meringue frosting thickly on the top and sides of the cake. Using a small metal spatula to create swoops and peaks.
Using a kitchen torch, lightly brown the frosting. Serve right away.