For the cake:
Lightly grease the bottom of a 9-inch round cake pan. Cover with a piece of parchment paper (or wax paper) cut to fit exactly. Lightly grease the paper and set the pan aside.
Preheat the oven to 350°F.
Place the cocoa in a small, heatproof bowl. Add 2 tablespoons of the sugar, mixing to blend. Gradually whisk in the milk, beating until blended. Set aside to cool.
Combine the flour, baking soda, and salt with the remaining sugar. Sift into a stainless-steel mixing bowl. Add the shortening, along with half of the cooled cocoa mixture, beating on low with a handheld electric mixer. Add the egg and vanilla along with the remaining cocoa mixture. Raise the speed to medium and beat to blend, scraping down the sides of the bowl with a rubber spatula from time to time.
Scrape the batter into the prepared cake pan, gently smoothing the top with a spatula. Bake for about 25 minutes, or until a cake tester inserted into the center comes out clean.
Remove the cake from the oven and place it on a wire rack to cool for 5 minutes. Then invert the pan onto the wire cake rack and tap it to release the cake from the pan. Peel off the paper and allow the cake to cool completely.
Either use the cooled cake immediately or wrap it in plastic film and refrigerate for up to 2 days or freeze until ready to use as directed in the recipe. This cake may also, of course, be iced and served as a simple cake.
For the baked alaska:
Line a 9-inch, 1½-quart stainless-steel mixing bowl with plastic film, leaving about a 3-inch overhang around the edge.
Using a rubber spatula, transfer the ice cream to the bowl, smoothing the top with the spatula. The entire bowl should be filled. Place in the freezer for about 3 hours or until completely solid.
Place the cake on a small baking sheet. Remove the ice cream from the freezer and invert the bowl onto it. Grabbing the plastic film that lines the bowl, lift the bowl from the ice cream and discard the plastic film. Again, return the dessert to the freezer and chill for about 1 hour, or until the cake is hard.
Preheat the oven to 400°F.
Combine the sugar and 1 cup water in a medium saucepan over medium-high heat. Attach a candy thermometer to the side of the pan and cook, stirring occasionally, for about 12 minutes, or until the mixture reaches the soft-ball stage, or 240°F on the thermometer.
Remove from the heat and keep warm.
Place the egg whites in the bowl of a standing electric mixer fitted with the wire whip and beat on medium-high until very foamy. Add the cream of tartar and salt, increase the speed to high, and beat until soft peaks form. Immediately begin pouring the hot syrup into the whites, allowing it to drizzle down the sides of the bowl to keep from splattering. Continue to beat on high until the mixture is cool and very shiny. (This is known as an Italian meringue.)
Remove the molded ice cream cake from the freezer, leaving it on the baking sheet. Using a rubber spatula, immediately begin spreading the meringue over the frozen ice cream, covering the entire dessert in a thick, even layer, shaping the meringue into soft peaks over all.
Once the dessert is covered, immediately bake it for about 8 minutes, or until the meringue is lightly browned and the ice cream is soft enough to be pierced with a metal cake tester or skewer.
Remove the Baked Alaska from the oven and carefully transfer it to a serving plate. Cut into slices as for a standard cake and serve.