1 1/2 sticks (6 ounces) unsalted butter, at room temperature
Confectioners' sugar, for sifting
1/2 cup granulated sugar
2 tablespoons water
Preheat the oven to 350°. Lightly coat two 12-by-17-inch jelly-roll pans with vegetable oil spray; line them with parchment or wax paper.
In a standing electric mixer fitted with a whisk, beat the egg whites until soft peaks form. At high speed, gradually add the granulated sugar and beat until the whites are glossy and firm, about 1 minute. Transfer the beaten whites to a large bowl; rinse and dry the mixing bowl.
Add the egg yolks, butter, confectioners' sugar, lemon zest, vanilla and salt to the mixing bowl and beat at moderate speed until creamy. Beat in one-fourth of the beaten whites. Transfer the batter to a large bowl. Fold in one–third of the flour, followed by one–third of the remaining whites; repeat with the remaining flour and whites until no streaks of white remain. Divide the batter between the prepared pans and spread evenly with an offset spatula. Bake the cakes for about 12 minutes, or until firm, shifting the pans halfway through baking.
In a large heatproof bowl set over a saucepan of simmering water, beat the eggs with the sugar until tripled in volume and an instant-read thermometer inserted in the mixture registers 165°, about 6 minutes. Whisk in the vanilla. Remove the bowl from the pan and pour out the water. Add the chocolate and water to the pan and cook over moderate heat, stirring, until the chocolate is melted. Scrape the chocolate into a medium bowl. Whisk in one-fourth of the egg mixture, then gradually whisk in the rest. Let cool.
In a large bowl, using a handheld mixer, beat the butter until fluffy. Beat in the chocolate mixture. Refrigerate until firm.
Liberally sift confectioners' sugar over the cakes. Top each jelly–roll pan with a baking sheet; invert to release the cakes. Remove the parchment and dust the cakes with confectioners' sugar. Cut each cake crosswise into three 5–by–12–inch rectangles. Place 1 cake layer on a platter. Beat the filling until thick and fluffy, about 2 minutes; spread 1/2 cup evenly over the layer. Repeat to make a total of 5 layers, reserving the last cake rectangle and 1/4 cup of filling. Thinly frost the side of the cake with the remaining 1/4 cup of filling. Refrigerate until very firm.
Spray a large sheet of wax paper with vegetable oil spray. Center the remaining cake layer on the sheet. In a medium saucepan, cook the sugar and water over moderately high heat, stirring, until an amber caramel forms, 5 minutes. Carefully pour the caramel over the cake layer and spread it thinly with a small offset spatula. (About one-fourth of the caramel will end up on the paper.) Working quickly, cut the cake in half crosswise with a buttered sharp knife, then cut it in half crosswise again to form 4 rectangles. Cut each rectangle into 3 strips to make 12 equal strips. Trim any excess caramel from the sides. Let the caramel harden completely.
Arrange the caramel cake strips side by side on the top of the cake. Cut into slices between the strips and serve.
The finished cake can be refrigerated, uncovered, for 2 days.