Preheat the oven to 350°F. With an electric hand mixer, beat the butter, brown sugar, and granulated sugar until light and fluffy. Then, add the eggs and whisk thoroughly.
Sift in the flour, baking powder, baking soda, cinnamon, and salt. Gently begin folding the batter together with a spatula, adding the milk halfway through. Once almost all of the flour has been absorbed, add the peeled and sliced apples. Fold them in evenly, and set the batter aside to prepare the crumble.
For the crumble, add the butter, flour, sugar, and cinnamon to a medium sized bowl. Pinch the ingredients together with your fingers until you achieve a crumbly texture.
Pour the cake batter into a greased and parchment lined 10” cake pan. The batter will be thick, so spread it to fill the pan evenly.
Top with the crumble and bake the cake for 40 minutes, or until the center of cake has fully set.
While the cake bakes, prepare the custard sauce. Add the milk and heavy cream to a saucepan over medium/low heat. Stir often to avoid burning.
In a small bowl, add the egg yolks and sugar. Whisk until smooth.
Once the cream mixture comes to a gentle simmer, begin tempering the eggs. Add one tablespoon of the hot cream to the egg yolk mixture, whisking constantly. Go VERY slowly to avoid scrambling the eggs. Add around a 1/2 cup of the hot cream to the egg mixture, then slowing pour the egg mixture into the sauce pan with the remaining cream. Again, mix constantly and pour slowly.
Allow the custard to cook over medium/low heat for another 5 minutes. Remove it from the heat and add the vanilla bean paste. Set aside.
Once the cake has cooled, remove it from the cake pan. Serve slices, topped with the custard sauce.