Line a 9-by-5-inch loaf pan with plastic wrap. Place the softened ice cream into the loaf pan and spread evenly. Place the cake on top of the ice cream, press down lightly, and place the pan in the freezer for 1 hour.
After 1 hour, combine the sugar, corn syrup, and 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 the egg whites, salt, and vanilla into the bowl of a stand mixer and, using the whisk attachment, whisk the eggs on high speed until they reach medium peaks, 4 to 5 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 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.
Remove the loaf pan from the freezer, turn upside down onto a heatproof serving platter, and remove the plastic wrap. Completely cover the ice cream cake with a 1-inch layer of meringue all around, sealing the meringue to the pan around the bottom edge.
Use a propane torch to brown the meringue all over. Serve immediately. May be frozen once the meringue has browned.