Preheat the oven to 350°F and grease two 9-inch pans with cake goop or line the bottom of the pan with parchment paper and coat with nonstick spray. Add 4 large egg whites to a stand mixer fitted with a whisk attachment and beat on speed 4 for 2 minutes, then increase to speed 6 until stiff peaks form, about 1 additional minute. Transfer egg whites to a small bowl and set aside.
In a separate small bowl or measuring cup, combine the 1 cup buttermilk and 1 teaspoon baking soda and set aside.
Add 1⅓ cups sweetened flaked coconut to a food processor and pulse until fine.
Add the processed coconut to the now empty stand mixer bowl
To the bowl with the coconut, add 2 cups cake flour, 2 cups granulated sugar, 2 tablespoons powdered milk, and ¾ teaspoon salt and mix to combine.
Add the softened 1 cup unsalted butter and mix until sandy in texture.
Scrape down the sides in case any butter is sticking and beat in 5 large egg yolks, 1 tablespoon vanilla bean paste, and 1½ teaspoons coconut extract.
Beat in the buttermilk mixture in three additions until fully combined, scraping down the sides as needed.
Fold in the stiff egg whites until well combined.
Evenly divide the batter between the two pans (about 720g each) and bake for 35 to 40 minutes until a toothpick comes clean.
Remove from oven and let cool in the pan for 5 minutes before transferring to a cooling rack until almost completely cool. These cakes are best made the day before frosting. When still a little warm to the touch, wrap them in a double layer of plastic wrap and store them at room temperature overnight - this helps make the cakes nice and moist.