Line an 11 x 9-inch quarter-sheet pan with parchment paper.
Combine the eggs, sugar, and salt in a stainless-steel bowl set over a saucepan of simmering water.
Lightly whisk until the temperature of the mixture reaches 110F.
This will help add volume to your sponge.
Transfer the egg mixture to the bowl of a stand mixer fitted with the wire whisk attachment (or use a hand mixer), and whip on high speed until very light, pale, and thick, 10 to 12 minutes.
I always set a timer for this stage because it is so important.
Carefully fold in flour, a quarter at a time, being careful not to deflate the foam.
Spread the batter into the prepared pan.
Bake the cake for 15 to 20 minutes, or until golden brown and the center springs back when touched.
Remove the pan from the oven.
Cool the cake in the pan completely before assembling.
(The cake can be made and frozen 2 weeks in advance.
Thaw before assembly.)
BAKE THE PASTRY
Preheat the oven to 400F.
Place the puff pastry sheets on a parchment paper-lined baking sheet.
Prick the pastry with a fork and brush with egg wash.
Bake the pastry for 15 to 20 minutes, or until golden brown, pricking the pastry with a fork every 5 minutes.
This will keep the pastry flat
MAKE THE PASTRY CREAM
In a medium saucepan over low heat, bring 1 cup milk and 1/2 cup sugar just to a boil.
In a mixing bowl, whisk cornstarch, egg yolks, remaining 1/4 cup sugar, and remaining 1 cup milk.
Slowly whisk hot milk mixture into egg mixture in a thin stream.
Return the mixture to the pan and whisk constantly over low heat until boiling and thick.
Remove from the heat and stir in the vanilla and butter.
Transfer half the mixture to a small bowl and place a piece of plastic wrap directly on the surface to prevent a skin from forming on top.
Cool completely before using.
Add the chocolate to the remaining half of the mixture and stir until the chocolate has melted and the mixture is smooth.
Transfer to a small bowl and place a piece of plastic wrap directly on the surface to prevent a skin from forming on top.
Cool completely before using.
(The pastry cream can be made 1 to 3 days in advance.
Place in an airtight container and refrigerate.)
MAKE THE RUM SYRUP
In a medium saucepan over medium heat, bring the sugar and water to a boil.
Boil until the sugar has dissolved, 3 to 4 minutes.
Remove from the heat and stir in the rum.
ASSEMBLE THE RUM SQUARES
Spread the chocolate pastry cream over one layer of puff pastry.
Top with the layer of sponge cake.
Brush and soak the sponge cake with rum syrup.
Spread a layer of vanilla pastry cream on top of the sponge cake and top with the remaining layer of puff pastry.