Preheat the oven to 350°F (175°C). In a large mixing bowl, combine the flour, sugar, baking powder, and salt. In another bowl, beat the eggs and mix in the milk, vanilla extract, and lemon juice. Gradually combine the wet and dry ingredients until smooth.

Bake the Cake

Pour the batter into a greased 9x13 inch baking pan. Bake for 25-30 minutes or until a toothpick inserted in the center comes out clean. Let it cool completely.

Soak the Cake

Once cooled, poke holes in the cake with a fork and soak it with additional lemon juice. Allow it to absorb for about 30 minutes.

Vanilla Cream Preparation

Make the Cream

In a mixing bowl, whip the heavy cream until soft peaks form. Gradually add the powdered sugar and vanilla bean paste, continuing to whip until stiff peaks form. Spread a layer over the soaked cake.

Lemon Sauce Preparation

Prepare the Sauce

In a small saucepan, combine lemon juice, sugar, cornstarch, and water. Cook over medium heat, stirring constantly until the mixture thickens. Let it cool slightly before drizzling over the cake.

Pistachio Crumble Preparation

Make the Crumble

In a food processor, pulse the pistachios until coarsely chopped. In a bowl, combine the chopped pistachios, butter, brown sugar, and flour. Mix until crumbly and spread on a baking sheet. Bake at 350°F (175°C) for 10-12 minutes until golden.

Assembly

Assemble the Cake

Once the cake is chilled and the toppings are prepared, layer the lemon sauce over the vanilla cream and sprinkle with pistachio crumble. Serve chilled.