Roll out three sheets of puff pastry, each to the same size and a thickness of 1/8 inch. Prick with a fork, and place on baking sheets.
Bake in preheated oven until golden, about 10 minutes.
In a medium bowl, combine cornstarch and 1/2 cup milk with fingers until smooth. Stir in cake flour and half the sugar. Beat in egg yolks. Set aside.
In a saucepan over medium heat, bring remaining milk and remaining sugar to a rolling boil. Stir in egg yolk mixture; continue stirring and bring to a boil again. Remove from heat and stir in butter and vanilla. Let cool.
In a small saucepan or in the microwave, heat the jam until runny.
Place one sheet of pastry on a board. Spread with cooled pastry cream to a thickness of 1/2 inch. Place a second pastry sheet over the cream. Brush with fruit glaze. Spread remaining pastry cream over glaze. Top with final pastry sheet and gently compress layers, using a baking sheet to press.
To make icing, beat together confectioners' sugar and lemon juice with enough of the egg whites to make a creamy consistency like that of a cream soup. Working quickly, color a small portion of the frosting with red food coloring. Frost the top and sides of the assembled pastry with white icing. Pipe parallel lines of red icing on top of dessert, then draw a pick or knife crosswise through icing to make the traditional design. Let icing set before cutting into bars.