Heat the oven to 350 °F and prepare three 6-inch or two 8-inch cake pans with butter, flour, and parchment circles on the bottoms. The parchment helps release the sticky cake without tearing its tender underside.

Combine the flour, sugar, cocoa, espresso powder, baking soda, baking powder, and salt in a large bowl, or use a stand mixer fitted with its paddle. An evenly colored dry mixture helps distribute the cocoa and leavening throughout the batter.

Add the egg, egg yolk, buttermilk, warm water, olive oil, and vanilla, then mix for about 1 minute. Stop when the batter looks smooth and fluid, without visible pockets of dry ingredients.

Portion the batter evenly among the prepared pans and bake for 27 to 30 minutes, checking the centers with a skewer. A clean skewer signals doneness, while wet batter means the layers need more time.

Leave the baked layers to cool and thoroughly wash and dry the mixing bowl before preparing the frosting. The cake must feel completely cool before filling, and the bowl must have no oily residue.

Whisk the water, sugar, egg whites, cream of tartar, and salt together in a large metal bowl, then position it over a pan of barely simmering water. The mixture will initially look loose and foamy rather than thick or spreadable.

Beat with a handheld electric mixer, starting on low and gradually increasing to high, for about 5 minutes until stiff peaks form. The foam should become glossy and dense enough that a lifted peak stands upright instead of folding over.

Move the bowl onto a folded kitchen towel on the counter, beat for about 2 minutes more, and mix in the vanilla. The finished frosting should feel cool and look billowy, with enough body to spread over the layers.

Combine the coconut and red food coloring in a bowl, then cover your hand with a clean zip-top bag and gently rub the flakes together. Keep mixing until the pink color looks even and no concentrated red patches remain.

Spread a few tablespoons of frosting from the center to the edge of the first cooled layer, add the second layer, and repeat before adding the third layer if making the 6-inch cake. Keep the rounds centered so the cake stands straight rather than leaning.

Cover the top and sides with the remaining frosting, smoothing it into an even coating. Small ridges are fine because the coconut will cover them.

Press coconut gently against the frosted sides, scatter it over the top, then slice and serve. The finished coating should cling to the frosting without being pressed deeply into it.