Preheat oven to 350°F (175°C) and grease two 8-inch pans, lining the bottoms with parchment.

Whisk the large egg, 5 egg whites, cream of coconut, water, vanilla extract, and coconut extract in a bowl until smooth.

Sift together the cake flour, granulated sugar, baking powder, and salt; cut in softened butter and coconut oil until the mix resembles coarse crumbs.

Fold the wet mixture into the dry mixture until just combined; the batter should be thick and glossy.

Divide the batter between the prepared pans, smooth the tops, and bake for 25–30 minutes or until a toothpick inserted comes out clean.

While the cakes bake, toast shredded coconut in a dry skillet over medium heat, stirring until golden and fragrant, about 3–5 minutes.

Whip the 4 egg whites with 1 cup sugar and a pinch of salt over a simmering water bath until thick and warm.

Beat in softened butter until silky smooth, then add the cream of coconut, coconut extract, and vanilla extract, whipping until light.

Cool the cake layers completely. Spread frosting between the layers, then cover the sides and top with frosting.

Press the toasted coconut onto the frosting and chill briefly to set.