Increase the oven temperature to 350°F (175°C). Grease and flour three 8-inch round cake pans; line bottoms with parchment rounds for easy release.
In the bowl of a stand mixer fitted with the paddle attachment, combine the flour, granulated sugar, baking powder, baking soda, and kosher salt. Mix briefly to combine.
With the mixer on medium speed, add the softened butter one piece at a time. Continue mixing until the mixture resembles coarse sand and the butter is fully incorporated into the dry ingredients.
In a large measuring cup, whisk together the egg yolks, buttermilk, vanilla extract, and almond extract until smooth.
With the mixer on low speed, slowly drizzle the buttermilk mixture into the flour mixture. Mix just until combined, scraping down the sides of the bowl as needed. Do not overmix.
Fold in the 1 ½ cups of roughly chopped toasted pecans and all of the toasted coconut.
In a clean, dry bowl using a hand mixer or the whisk attachment of a stand mixer, beat the egg whites with the cream of tartar on medium-high speed until stiff, glossy peaks form (3 to 4 minutes).
Gently fold the whipped egg whites into the batter in two to three additions using a spatula. Use light strokes and fold only until no white streaks remain.
Divide batter evenly into pans. Bake on center rack 28 to 32 minutes (checking at 25 minutes). A toothpick should come out with a few moist crumbs, but no wet batter. If the oven won't fit all three, bake in batches or rotate halfway.
Allow the cakes to cool in the pans for 10 to 15 minutes, then invert onto racks to cool completely.