Cake:Preheat oven to 350 degrees F. Spray a 9×13 pan with non-stick spray.
In an electric mixer, cream butter and sugar until fluffy. Add in applesauce and eggs. Mix in buttermilk and vanilla.
Add flour, baking powder and salt and mix until combined. Turn mixer to high for 5 seconds to make sure ingredients are incorporated.
Pour into prepared pan and bake for 28-30 minutes or until a toothpick comes out clean. Set aside to cool slightly while you prepare the pudding.
Pudding:In a large pot, combine milk, egg, sugar, corn starch, and salt with a whisk until smooth. Cook over medium heat, whisking constantly, until thickened (so that it coats the back of a spoon like a thin pudding -- will thicken slightly as it cools). Stir in extracts.
Assembly:Using a wooden spoon, poke holes 1″ apart in the slightly cooled cake. Pour warm pudding over.
Let set in the fridge until cold, at least 2-3 hours.