Instructions

1. Place 2 sticks unsalted butter, 4 large eggs, and 1 cup whole milk on the counter and let sit at room temperature until the butter is softened.
2. Arrange a rack in the middle of the oven and heat the oven to 350°F. Coat a 9x13-inch baking dish with cooking spray.
3. Place 1 cup of the granulated sugar and 1 cup of the shredded coconut in a food processor fitted with the blade attachment. Process until the coconut is very finely ground with the sugar, 15 to 20 seconds. Transfer to the bowl of a stand mixer or a large bowl if using an electric hand mixer.
4. Add the butter and 1 cup of the granulated sugar to the coconut sugar and beat with the paddle attachment on medium-high speed until very creamy and soft, scraping down the sides halfway through mixing, 4 to 5 minutes total. Scrape down the sides of the bowl with a spatula. Add the eggs one at a time, beating well after each addition. Beat in 2 teaspoons of the vanilla extract.
5. Place 3 cups all-purpose flour, 1 tablespoon baking powder, and 1 1/4 teaspoons kosher salt in a large bowl and whisk to combine.
6. With the mixer on low speed, beat in 1/3 of the flour mixture, then beat in 1/2 of the milk. Beat in 1/2 of the remaining flour mixture, then the remaining milk. Beat in the remaining flour mixture and continue beating just until smooth and combined. Use a spatula to scrape down the sides of the bowl and to make sure all the ingredients are evenly combined. Scrape the batter into the baking dish and spread into an even layer. spreading it evenly.
7. Bake until the cake is firm and golden brown, and a toothpick inserted into the center of the cake comes out with just a few moist crumbs clinging to it, 30 to 35 minutes. Meanwhile, place 3/4 cup of the heavy cream, 1 can sweetened condensed milk, 1 cup evaporated milk, 3/4 cup cream of coconut, and 1/4 cup dark rum in a medium bowl and stir to combine.
8. Transfer the cake to a wire rack and let cool 15 minutes (keep the oven on). Meanwhile, sprinkle the remaining 1/2 cup shredded coconut on a rimmed baking sheet. Bake, stirring every 3 minutes, until toasted and golden-brown, about 6 minutes total. Cool completely and store in an airtight container until ready to use.
9. While the cake is still warm, use a skewer to poke holes all over the top of the cake, poking all the way to the bottom of the cake. Ladle half of the milk syrup over the cake, then wait for it to soak in, about 5 minutes. Pour the remaining milk syrup over the cake, using it all. Let the cake cool completely. Cover in plastic wrap and refrigerate at least 4 hours but preferably overnight.
10. Top the cake right before serving: Place the remaining 2 cups cold heavy cream, remaining 1 teaspoon vanilla extract, and remaining 1/4 cup granulated sugar in the bowl of a food processor fitted with the blade attachment. Pulse until the cream is dense, thickened, and forms soft peaks, 3 to 4 (15-second) pulses. Do not over-whip or the cream will turn to butter. Spread on the cake and sprinkle with the toasted coconut. Cut into squares and serve. This cake is best served cold.