In a medium saucepan, heat 2 cups of whole milk over medium heat until just simmering. Whisk together 6 egg yolks, 3/4 cup granulated sugar, a pinch of salt, 1 tablespoon vanilla bean paste, and 3 tablespoons of cornstarch in a separate bowl. Gradually combine the heated milk into the egg mixture, stirring continuously. Return to the saucepan and cook over medium heat, whisking constantly until thickened, about 5-7 minutes. Chill in the fridge for at least 2 hours.

Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking pan, lining it with parchment paper. In a large mixing bowl, whisk together 2 cups of all-purpose flour, 1.5 teaspoons of baking powder, and 0.5 teaspoon of baking soda. In another bowl, cream 1 cup of unsalted butter and 1 cup of granulated sugar until light and fluffy, about 3-4 minutes. Add in 4 eggs and 1 tablespoon of vanilla bean paste, and mix until fully incorporated.

Gradually mix the whisked dry ingredients into the creamed mixture, alternating with 1 cup of buttermilk, starting and ending with dry ingredients. Mix until just combined to avoid overworking the batter.

Pour the batter into the prepared pan, spreading it evenly. Bake for 39-44 minutes, or until a toothpick inserted into the center comes out clean and the top is golden brown. Allow the cake to cool in the pan for 30 minutes before transferring to a wire rack.

In a bowl, combine 1 cup of half & half and 1 cup of sweetened condensed milk along with 1 tablespoon of vanilla bean paste. Whisk until smooth and chill in the refrigerator.

Poke holes all over the cooled cake using a fork. Pour the chilled milk mixture evenly over the cake, allowing it to absorb for about 15 minutes. Spread the chilled pastry cream over the top of the cake.

Sprinkle an even layer of granulated sugar over the pastry cream. Using a kitchen torch, carefully caramelize the sugar until golden brown.