Center a rack in the oven and preheat to 350 (F). Prepare four 8-inch cake pans by spraying generously with cooking spray. Line the bottom of each cake pan with a parchment paper circle and spray with cooking spray as well. Set pans aside.
In a medium bowl, combine 3 cups cake flour, 3 teaspoons baking powder, and 1 teaspoon kosher salt. Whisk to combine and set aside.
Place 5 large egg whites in a large bowl and use a handheld electric hand mixer to whisk on medium speed until the egg whites are frothy. Raise the mixer speed to high and gradually beat in 1/4 cup granulated sugar. Continue whisking until the egg whites form stiff peaks. Set aside. In the bowl of a freestanding electric mixer fitted with a paddle attachment, combine the remaining 2 cups granulated sugar and 6 tablespoons unsalted butter. Beat on medium speed until light and fluffy, about 3 to 5 minutes. Lower the mixer speed to its slowest setting and add 5 large egg yolks, one at a time, only adding the next egg yolk when the previous one is fully incorporated. Once all the egg yolks have been added, add 1 cup ube halaya jam, 1 tablespoon ube extract, and 2 teaspoons purple food coloring. Continue beating until combined and the mixture is a homogenous bright purple, using a rubber spatula to scrape down the paddle and the sides and bottom of the bowl as needed.
With the mixer on low speed, add the dry ingredients (from the 2nd step) in three parts, alternating with 1 1/4 cups whole milk in two parts, beginning and ending with the dry ingredients. Mix on low speed just until the ingredients are combined.
Once the ingredients are combined, remove the bowl from the mixer and fold in one-fourth of the beaten egg whites (from the 3rd step) to lighten the batter. Once incorporated, fold in the rest of the egg whites until no steaks remain — be careful not to fold or mix too much, or the batter will deflate and you'll end up with flat and dense cakes. Fold only until the whites disappear into the batter. The batter should lighten and look almost like a purple mousse.
Divide the batter into the 4 prepared cake pans and transfer to the preheated oven. Bake for 30 to 35 minutes, or until a skewer inserted into the center of each cake comes out clean and the top of each cake bounces back when gently poked. Transfer to a wire rack to cool completely before decorating.