For the ice cream cake: Butter a 9-inch springform pan and line it with a parchment paper circle. Cover the bottom of the pan with the brownies, using any extra pieces to line the sides of the pan if desired. Sprinkle with the cookies. Scoop the ice cream onto the cookie layer, packing it in tightly. Spread with an offset spatula to make a smooth 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 unhinging the springform ring and removing it. 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 vanilla whipped cream frosting: Add the cream, confectioners' sugar and vanilla to a large bowl and beat 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 decorate with the sprinkles. If the cake begins to get too soft, put it back into the freezer for a few minutes.
Freeze an additional 1 1/2 hours and up to overnight. Before serving, let the cake sit at room temperature for about 20 minutes.