Heat the oven to 350ºF. Lightly oil and flour a 9-by-13-inch metal baking pan and set aside.
Whisk together the cake flour, baking powder, and salt in a medium mixing bowl and set aside.
Put the butter in the bowl of a stand mixer fitted with the paddle attachment and cream on medium speed until fluffy, approximately 1 minute. Decrease speed to low and, with the mixer still running, gradually add the sugar over 1 minute. Stop to scrape down the sides of the bowl if necessary.
Add the eggs one at a time and mix to thoroughly combine. Next, add the vanilla and mix to combine. Add the flour mixture to the batter in 3 batches and mix just until combined.
Spread the batter evenly in the prepared pan — it will appear to be a very small amount of batter. Bake on the middle rack of the oven until the cake is lightly golden and reaches an internal temperature of 200ºF, 20 to 25 minutes.
Remove the pan to a cooling rack and cool for 30 minutes. Poke the top of the cake all over with a skewer or fork.
Make the glaze: Whisk together the evaporated milk, sweetened condensed milk, and half-and-half in a 1-quart measuring cup or other large pour-friendly vessel. Pour the glaze over the cake. Refrigerate the cake overnight.
Make the topping: Put the cream, sugar, and vanilla in the bowl of a stand mixer fitted with the whisk attachment and whisk on low speed until it forms soft peaks. Increase the speed to medium and whip until thick. Spread the topping over the cake and chill until ready to serve.