Baked Alaska

Lay 12-inch square sheet of plastic wrap on counter and remove lids from ice cream. Use scissors to cut cardboard tubs from top to bottom. Peel away cardboard and discard. Place ice cream blocks on their sides in center of plastic with wider ends facing each other. Grasp each side of plastic and firmly press blocks together to form barrel shape. Wrap plastic tightly around ice cream and roll briefly on counter to form uniform cylinder. Place cylinder, standing on end, in freezer until completely solid, at least 1 hour.

FOR THE CAKE: Adjust oven rack to middle position and heat oven to 350 degrees. Lightly grease 18 by 13-inch rimmed baking sheet, line with parchment paper, and lightly grease parchment. Whisk flour, cocoa, 1/3 cup sugar, baking powder, and salt together in large bowl. Whisk oil, water, and egg yolks into flour mixture until smooth batter forms.

Using stand mixer fitted with whisk attachment, whip egg whites on medium-low speed until foamy, about 1 minute. Increase speed to medium high and whip whites to soft, billowy mounds, about 1 minute. Gradually add remaining 1/3 cup sugar and whip until glossy, soft peaks form, 1 to 2 minutes. Transfer one-third of egg whites to batter; whisk gently until mixture is lightened. Using rubber spatula, gently fold remaining egg whites into batter.

Pour batter into prepared sheet; spread evenly. Bake until cake springs back when pressed lightly in center, 10 to 13 minutes. Transfer cake to wire rack and let cool for 5 minutes. Run knife around edge of sheet, then invert cake onto wire rack. Carefully remove parchment, then reinvert cake onto second wire rack. Let cool completely, at least 15 minutes.

Transfer cake to cutting board with long side of rectangle parallel to edge of counter. Using serrated knife, trim 1/4 inch off left side of cake and discard. Using ruler, measure 4 1/2 inches from cut edge and make mark with knife. Using mark as guide, cut 4 1/2-inch rectangle from cake. Trim piece to create 4 1/2 by 11-inch rectangle and set aside. (Depending on pan size and how much cake has shrunk during baking, it may not be necessary to trim piece to measure 11 inches.) Measure 4 inches from new cut edge and make mark. Using mark as guide, cut 4-inch rectangle from cake. Trim piece to create 4 by 10-inch rectangle, wrap rectangle in plastic, and set aside. Cut 3 1/2-inch round from remaining cake and set aside (biscuit cutter works well). Save scraps for Bonus Baked Alaska.

Unwrap ice cream. Trim cylinder to 4 1/2 inches in length and return remainder to freezer for Bonus Baked Alaska. Place ice cream cylinder on 4 1/2 by 11-inch cake rectangle and wrap cake around ice cream. (Cake may crack slightly.) Place cake circle on one end of cylinder. Wrap entire cylinder tightly in plastic. Place cylinder, standing on cake-covered end, in freezer until cake is firm, at least 30 minutes.

Unwrap cylinder and place on cutting board, standing on cake-covered end, and cut in half lengthwise. Unwrap reserved 4 by 10-inch cake rectangle and place halves on top, ice cream side down, with open ends meeting in middle. Wrap tightly with plastic and press ends gently to close gap between halves. Return to freezer for at least 2 hours and up to 2 weeks.

FOR THE MERINGUE: Adjust oven rack to upper-middle position and heat oven to 500 degrees. Spray wire rack set in rimmed baking sheet with vegetable oil spray. Unwrap cake and place on rack. Combine sugar, corn syrup, egg whites, water, and salt in bowl of stand mixer; place bowl over saucepan filled with 1 inch simmering water, making sure that water does not touch bottom of bowl. Whisking gently but constantly, heat until sugar is dissolved and mixture registers 160 degrees, 5 to 8 minutes.

Place bowl in stand mixer fitted with whisk attachment. Beat mixture on medium speed until bowl is only slightly warm to touch, about 5 minutes. Increase speed to high and beat until mixture begins to lose its gloss and forms stiff peaks, about 5 minutes. Add vanilla and beat until combined.

Using offset spatula, spread meringue over top and sides of cake, avoiding getting meringue on rack. Use back of spoon to create peaks all over meringue.

Bake until browned and crisp, about 5 minutes. Run offset spatula or thin knife under dessert to loosen from rack, then use two spatulas to transfer to serving platter. To slice, dip sharp knife in very hot water and wipe dry after each cut. Serve immediately.