Preheat your oven to 350°F (175°C) and grease a 7-inch square cake pan, then line it with parchment paper. In a small saucepan, gently melt the butter and set it aside to cool slightly. In a large bowl, whisk together the eggs and sugar at high speed until the mixture becomes pale and mousse-like. It should be thick enough that when you lift the whisk, a trail remains in the mixture.

Sift the flour, cornstarch, baking powder, and salt into a bowl. Gently fold half of this dry mixture into the egg mixture. Slowly pour half of the cooled butter around the edge of the batter, then fold it in carefully. Repeat with the remaining flour mixture and butter, making sure not to deflate the batter.

Pour the batter into the prepared pan, leveling the surface with a spatula. Bake for 35–40 minutes, or until the cake has risen and springs back when lightly pressed. Once baked, let it cool in the pan for a few minutes before turning it out onto a wire rack. Peel off the parchment paper and allow it to cool completely.

In a small saucepan, combine the sugar and water. Heat gently until the sugar dissolves, then bring to a boil. Boil for about 5 minutes, or until the syrup is clear and forms a thin thread when pulled between two teaspoons. Meanwhile, place the egg yolk in a bowl and stir it lightly. Slowly pour the syrup over the egg yolk in a thin stream, whisking continuously. Keep whisking until the mixture thickens and cools. In a separate bowl, cream the softened butter, then gradually mix in the cooled egg mixture. Stir in the coffee to flavor the buttercream.

Once the cake has cooled, slice it in half horizontally using a serrated knife. Spread a thin layer of the coffee buttercream between the two layers, then press them back together. Trim the edges of the cake to create a neat finish, then cut the cake into 8 oblong pieces.

Sift the apricot jam into a small saucepan and heat it gently until warmed through. Brush the tops and sides of each cake with the hot apricot jam to give it a glossy finish.

In a small saucepan, heat the butter, milk, and coffee together until the butter melts. Add the sifted confectioners’ sugar and beat until smooth and glossy. Let the frosting thicken slightly before using most of it to pour over each cake. Smooth the sides of the cakes quickly if needed. Allow the frosting to set, then use the remaining frosting to pipe decorative accents on top.