Line two 13-by-9-inch metal cake pans with plastic wrap and place in the freezer.
Pour the chocolate ice cream base into the bowl of a stand mixer fitted with the paddle attachment. Turn the speed to low. With the mixer running, slowly pour in the liquid nitrogen until the ice cream reaches a classic soft-serve consistency.
Spread the ice cream evenly in one of the prepared pans and freeze until firm, at least 2 hours. May be made several days in advance.
Clean the bowl and paddle and repeat with the strawberry ice cream.
Set the chiffon cake on a parchment-lined half-sheet pan and place in the freezer for at least 1 hour and up to several hours in advance of assembling the Baked Alaska.
Combine 8 ounces sugar, 4 ounces corn syrup, and 2 fluid ounces water in a 2-quart saucepan and place over high heat. Stir just until the sugar dissolves, about 5 minutes. Clip on a candy thermometer and bring the mixture to 240ºF.
While the syrup is cooking, place 4 ounces egg whites, 1/2 teaspoon vanilla, and 1/8 teaspoon salt into the bowl of a stand mixer fitted with the whisk attachment and whisk the eggs on high speed until they reach medium peaks, 2 to 3 minutes.
When the sugar syrup reaches 240ºF, remove from the heat and, with the stand mixer on low speed, slowly and carefully pour the syrup in a thin, steady stream into the egg white mixture, being careful not to come in contact with the whisk. Once all of the syrup has been added, increase the mixer speed to high and whisk until stiff peaks are formed and the mixture has cooled, 8 to 10 minutes. Transfer the mixture to a large piping bag and refrigerate while preparing the second and third batches. Repeat the recipe two more times for a total of three batches of meringue.
Remove the meringue from the refrigerator 30 minutes prior to using.
To assemble the Baked Alaska: Cover the perimeter of a large (slighly larger than the cake), wooden cutting board with strips of heavy-duty aluminum foil. Remove the cake and ice cream from the freezer. Place the cake on the prepared board. Invert the sheet pans of ice cream atop the cake. Peel away the plastic wrap from the ice cream. Pipe the meringue over the entire surface of cake and ice cream, covering completely.
Use a propane torch to brown the meringue. Serve immediately or freeze for up to 1 week once the meringue has been browned.