Butter a 9-inch springform pan and line the bottom with a parchment paper circle.
Cover the bottom of the pan with the pound cake, using any extra pieces to line the sides of the pan if desired. Scoop the chocolate ice cream onto the cake layer. Spread with an offset spatula to make a smooth top. Scoop the strawberry ice cream over the chocolate layer and smooth the top. Scoop the pistachio ice cream over the strawberry layer and smooth the top. Cover with plastic wrap and freeze until firm, 2 to 2 1/2 hours.
Unmold the cake by running a knife around the edge of it, then unhinge the springform and remove. Invert the cake onto a sheet of plastic wrap and remove the bottom of the pan and the parchment. Place a plate on the bottom and invert once more.
For the chocolate whipped cream frosting:
Put the heavy cream, chocolate syrup, confectioners' sugar, and vanilla in a large bowl and whip with an electric mixer until soft peaks form. (This makes enough frosting to cover the top and sides of the cake with additional to pipe decorations if desired.)
Frost the top and sides of the cake with the whipped cream frosting and sprinkle with the pistachios. If the cake begins to get too soft, put it back into the freezer for a few minutes.
Freeze another 1 1/2 to 2 hours or overnight. Before serving, let the cake sit at room temperature for about 20 minutes.
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