To make the strawberry sauce, puree the strawberries and sugar in a food processor until liquefied. Lighty coat a 10 1/2 by 15 1/2-inch jelly roll pan with nonstick vegetable spray. Add the batter to the pan and bake for 13 to 15 minutes, until lightly tanned and springy. Set on a rack to cool. Cut out two circles of cake, one 5 inches in diameter; the other 4 1/4 inches in diameter. Cut out a rectangle 3 by 15 1/2 inches. Sprinkle the cake with the orange liqueur. Wrap the ice cream container in a hot towel to loosen it. Invert the container onto the larger cake circle. Top with the smaller circle. Use the cake rectangle to wrap the sides, making sure that the wet side of the cake is facing in. Fill any gap with left over scraps. Return the cake to the freezer. Set a rack towards the bottom of the oven with plenty of room above it. Preheat the oven to 450 degrees. To make the Italian Meringue, put the water, corn syrup, and sugar in a small saucepan fitted with a candy thermometer. Over high heat, bring the temperature to 246 degrees. In the meantime, completely clean and dry mixing bowl. With an electric mixer, whisk the egg whites, salt, and cream of tartar until creamy, foamy, and barely able to hold very soft peaks. With the mixer running on slow, carefully drizzle in the hot syrup. Continue to whisk on high, for about 2 minutes, until the consistency of shaving cream; slightly glossy and able to hold soft peaks. Set the cake on a heatproof serving platter. With a metal spatula, spread the meringue entirely around the cake. With a pastry bag fitted with a star tip, pipe any leftover meringue in a decorative pattern. Pipe one large rosette in the top and plant the eggshell in it. Dust the meringue with the confectioners' sugar. Bake for 2 to 3 minutes until lightly tanned. Pour the strawberry sauce around the bottom of the platter. In the meantime, gently warm the rum or brandy in a small saucepan. Pour it into the egg shell and drizzle the rest over the cake. Ignite!
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