Step 1

Preheat oven to 400°. Line the bottom of a 13x9x2" metal baking pan with parchment paper. Butter paper and sides of pan; dust with cocoa powder, tapping out excess. Sift flour and remaining 1/4 cup cocoa powder through a fine-mesh sieve into a small bowl. Repeat sifting 2 more times. Set flour mixture aside.

Step 2

Crack 2 eggs into a large deep bowl. Separate remaining 2 eggs, adding yolks to bowl with whole eggs and placing whites in a medium deep bowl. Set whites aside.

Step 3

Using an electric mixer, beat whole eggs and yolks on low speed for 1 minute. Increase speed to medium; gradually add 1/2 cup sugar. Beat until mixture is thick and pale, about 3 minutes. Beat in vanilla.

Step 4

Using clean, dry beaters, beat egg whites until foamy. Sprinkle 1 tsp. sugar over. Continue to beat until soft peaks form, about 1 minute. Sprinkle remaining 5 tsp. sugar over and beat until meringue is thick and glossy, about 30 seconds.

Step 5

Using a rubber spatula, fold half of meringue into egg mixture. Sift half of dry ingredients over batter and fold until just blended. Repeat with remaining meringue and dry ingredients.

Step 6

Scrape batter into prepared pan, spreading evenly to edges. Bake until cake springs back when pressed gently with your fingertips, 10–12 minutes.

Step 7

Let cake cool in pan on a wire rack. Run a knife around sides of pan to release cake.

Step 8

Place a cutting board on top of pan. Invert cake onto board; remove pan. Carefully peel away parchment. Using a serrated knife, trim edges to create an even layer.

Step 9

Bring cream to a simmer in a small saucepan over medium-low heat. Remove from heat; add chocolate chips and stir until icing is smooth and glossy. Let cool to thicken slightly, about 15 minutes.

Step 10

Using an offset spatula, spread half of icing down the middle of the cake; smooth evenly to the edges. Cut cake crosswise into 3 pieces. Stack pieces to make 3 layers. Spread remaining icing on sides of cake. DO AHEAD: Can be made 2 days ahead. Store airtight at room temperature.