
Combine ½ cup of water with ½ cup of granulated sugar in a small saucepan, stirring continuously until the sugar has dissolved.
In a small bowl, dissolve 1 teaspoon of espresso powder in 1 tablespoon of water, and set aside. Let the cake cool, then run a knife around the edges of the pan to help loosen the cake from the pan. Bake for 8 to 10 minutes, until the cake is lightly golden and springs back when you touch it. Pour this completed cake batter into the prepared sheet pan, spreading it evenly. Melt 2 tablespoons of butter and gently fold that into the cake mixture. Gently fold the egg white mixture into the cake mixture. In a separate bowl, use an electric mixer to beat the 3 egg whites with ¼ cup of granulated sugar, until stiff peaks form. In a large bowl, make the cake mixture by beating the almond flour, all-purpose flour, confectioner's sugar, 1 egg yolk, and the remaining 3 whole eggs, until well combined.
Separate 3 eggs into 3 bowls as follows: all egg whites in the first bowl, 2 yolks in a second bowl, and the third yolk in a third bowl.
Grease an 18×13-inch sheet pan and then line it with parchment paper or a Silpat mat, and lightly grease that as well.