Place oven racks in the upper and lower thirds of the oven and heat to 350°F (175°C). Grease two 9-by-13-inch metal baking pans, line their bottoms with parchment paper, and lightly grease the parchment. If you have only one pan, bake the layers one after the other.

Separate the eggs, placing whites in a clean large bowl and yolks in a second large bowl. Beat the egg whites with an electric mixer on medium-high speed until foamy. Gradually beat in 1/3 cup (67 g) of the sugar and continue until glossy medium peaks form. Set aside.

Add the remaining 1/3 cup (68 g) sugar and vanilla to the yolks. Beat on medium-high speed for 3 to 4 minutes, until the mixture is pale, thick, and falls from the beaters in ribbons.

Whisk together the flour, baking powder, and salt. Fold one-third of the whipped whites into the yolk mixture to lighten it. Fold in the dry ingredients in two additions, alternating with the melted butter and milk. Gently fold in the remaining whites just until no large white streaks remain.

Divide the batter evenly between the prepared pans and spread it into thin, even layers. Bake for 8 to 10 minutes, rotating pans halfway through, until the cakes spring back lightly when touched and the edges are pale golden. Cool in the pans for 5 minutes, then turn out onto racks, peel away the parchment, and cool completely.

Make the buttercream while the cakes cool. Beat the softened butter until smooth and creamy. Beat in the powdered sugar and cocoa powder in small additions, starting on low speed. Add the cooled melted chocolate, milk or cream, vanilla, and salt. Beat until fluffy. Add a little more milk only if needed for a smooth, spreadable consistency.

Trim any dry cake edges. Cut each cooled 9-by-13-inch cake lengthwise into three equal strips, making six sponge layers about 4 by 9 inches. Choose the straightest strip for the caramel top and set it aside.

Place one sponge strip on a serving board or platter. Spread a thin, even layer of buttercream over it. Repeat with four more sponge strips, stacking carefully and keeping the sides aligned. Set the final sponge strip aside for the caramel. Use any remaining buttercream to lightly coat the sides of the stack. Chill the cake while you make the topping.

For the caramel, place the sugar, water, and lemon juice in a small light-colored saucepan. Cook over medium heat without stirring until the sugar dissolves, then gently swirl the pan as the syrup turns a deep amber color, about 7 to 10 minutes. Remove from the heat immediately.

Place the reserved sponge strip on parchment paper. Carefully pour the hot caramel over it and tilt the strip to coat the surface. Lightly oil a sharp knife. Working quickly before the caramel hardens, score the topping into 12 portions or cut it into decorative angled pieces. Let it cool completely and harden.

Arrange the caramel-topped pieces over the chilled cake just before serving. For cleaner slices, use a sharp knife dipped in hot water and dried between cuts.