Heat 2 cups of whole milk in a saucepan over medium heat until steaming but not boiling. In a separate bowl, whisk 5 egg yolks, 1 cup of granulated sugar, a pinch of salt, 1 tablespoon of vanilla bean paste, and 2 tablespoons of cornstarch. Slowly mix hot milk into the egg mixture, then return it to saucepan on low heat, stirring constantly until thickened, about 5–7 minutes. Transfer to a bowl, cover, and chill for at least 2 hours.

Preheat oven to 350°F (175°C) and grease a 9x13 inch baking pan. In a bowl, combine 2 cups of all-purpose flour, 1 teaspoon of baking powder, and ½ teaspoon of baking soda. In another bowl, cream together ½ cup of softened unsalted butter and 1 cup of sugar until fluffy. Add 3 eggs and 2 teaspoons of vanilla extract, mixing well. Gradually add the dry ingredients and 1 cup of buttermilk, mixing until just combined.

Pour cake batter into greased pan, spreading evenly. Bake in preheated oven for 39-44 minutes, until a toothpick comes out clean. Allow cake to cool in pan for 15 minutes before transferring to a wire rack.

Whisk together 1 cup of half & half, ½ cup of sweetened condensed milk, and 1 tablespoon of vanilla bean paste. After cake has cooled, trim the top to level it. Poke holes in the surface to help milk soak absorb. Pour milk soak evenly over the top and let it absorb for about 30 minutes.

Spread chilled pastry cream over the soaked cake. Cut into squares or rectangles while still in the pan and refrigerate for another 30 minutes.

Sprinkle an even layer of granulated sugar over the pastry cream. Use a kitchen torch to caramelize sugar until it turns golden brown, creating a crispy crust.