Grease and lay parchment paper over the bottom of a jelly roll size baking pan (17 1/2 by 11 1/2-inches by 1-inch deep). With an electric beater, beat together the egg yolks, just 1/4 cup of the sugar (reserving the rest), and vanilla until the mixture becomes a pale yellow and thickens. Sift cake flour over the egg yolk mixture and set aside briefly. Wash and dry the beaters. In a separate metal bowl, beat the egg whites and cream of tartar together and gradually beat in the balance of the sugar until stiff peaks form. To the egg yolk/flour bowl, fold in 1/3 of the beaten egg whites until just barely combined. Repeat 2 more times with each of the remaining 1/3 of the beaten egg whites. Spread into jelly roll pan and bake in oven for about 15 minutes or until the surface is golden brown and springs back when touched. Let cake cool in pan.
While the cake is cooling, it would be a good idea to make room in the freezer for the sheet cake and ice cream.
When the cake is completely cool, peel off the browned surface of the cake. (You can do this simply by loosening it and rolling it off.) Firmly press a flat baking sheet without edges (which is the same size or larger than the baking tin) against the top edges of the baking tin and invert the cake so it is transferred to the flat sheet and remove the parchment paper. Then cut the cake in half lengthwise and in half crosswise. Two of these quarters will be dedicated to the top and bottom of the baked Alaska, respectively. Using a broad thin spatula, slide one of these quarters to a flat oven safe platter.
Loosen the carton from the half-gallon of ice cream and place the entire brick onto the cake on the platter. Then slide the other quarter onto the top of the brick of ice cream and put the whole platter into the freezer.
The remaining 2 pieces of cake will fit over the sides and ends of the brick of ice cream. Cut the cake to fit the dimensions of the sides and ends utilizing the most efficient positioning which will result in the fewest number of cuts. You will need 2 rectangles measuring approximately 8 inches by 3 inches and 2 rectangles measuring approximately 8 by 5-inches. Cover with plastic wrap if you are not using these immediately, to prevent them from drying out.
Some of the meringue topping will be used to secure these rectangles to the sides of the ice cream.
To make the meringue topping, make a slurry with 2/3 cup water gradually whisked into a small saucepan to which you have added cornstarch and sugar. Bring to a boil over medium heat, whisking constantly. As it thickens and becomes translucent, cover and remove it from the heat. In a scrupulously clean large metal or glass bowl, with scrupulously clean beaters, beat the egg whites (they must be at room temperature) until foamy. Add vanilla and cream of tartar and beat until soft peaks form. Very gradually beat in sugar on high until glossy and stiff peaks form, but do not allow it to get dry. Reduce the speed of the beater to low and gradually add the cornstarch slurry. When it has all been added increase speed to medium for about 10 seconds.
This can be refrigerated with plastic wrap on top for a short while before use.
Immediately before serving the Baked Alaska, make sure your oven rack is in a position to provide enough clearance for the meringue and pre-heat oven to 425 degrees F. Remove the sheet cake and ice cream from the freezer. Using some of the meringue as a "glue" position the pre-cut pieces of cake to fit the sides and the ends of the brick of ice cream. Then spread the rest of the meringue in a thick layer over the entire assembly, swirling it decoratively with the back of a spoon, and being sure to cover the lower edge of the cake on all sides so the meringue touches the pan base. Slide the Alaska into the oven and bake until the meringue browns, 5 or 10 minutes. Serve immediately with accompaniments of chocolate sauce, raspberry sauce, and caramel sauce.