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For the cake: position a rack in the middle of the oven.
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Preheat the oven to 350F.
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Coat two 9 x 2-inch round cake pans with nonstick spray, line the bottoms with parchment rounds, and then spray the parchment.
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1) Whisk together the flour, baking powder, and salt in a medium bowl to aerate and combine; set aside.
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2) In the bowl of stand mixer, add the butter and beat with the flat paddle attachment on medium-high speed until creamy, about 2 minutes.
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Add the sugar gradually and continue to beat until very light and fluffy, about 3 minutes more.
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Beat in the vanilla.
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Add the eggs one at a time, beating well after each addition.
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Add the flour mixture in three additions, alternately with the coconut milk.
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Begin and end with the flour mixture; add 1 cup of the coconut and beat briefly until smooth.
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Divide the batter evenly in the pans and smooth the tops with an offset spatula.
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Reserving 1 1/4 cups of coconut, spread the remaining 1 1/4 cups coconut out in a single layer on a jelly-roll pan.
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3) Toast the coconut in the oven, shaking the pan once or twice, until it is golden, about 5 minutes.
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Put the cake layers in the oven at the same time and bake for 20 to 25 minutes, or until a toothpick shows a few moist crumbs when removed.
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The cakes will be tinged light golden brown and will have begun to come away from the sides of the pans.
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When the cakes are done, place the pans on racks and immediately poke the cakes all over with a bamboo skewer.
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Brush the cakes evenly and equally with half of the rum.
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Cool the cakes completely in the pans on racks.
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Unmold, peel off the parchment, and brush this bottom side (now on top) with the remaining rum.
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Trim the layers to be level, if necessary.
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The layers are ready to fill and frost.
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Alternatively, place the layers on cardboards and double-wrap in plastic wrap; store at room temperature and assemble within 24 hours.
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Cool the toasted coconut in the pan on a rack as well.
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Set this toasted coconut aside for decorating the outside of the cake.
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For the Lime Cream: put the sugar, juice, and yolks in the top of a double boiler.
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Whisk together to break up the eggs.
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Add the butter.
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Place over the bottom of a double boiler filled with hot water that is just touching the bottom of the bowl.
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Place over medium heat and bring the water to a simmer.
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1) Whisk the mixture frequently over simmering water for about 12 minutes, or until the mixture reaches 180F.
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(The temperature is more important than the time it takes, and the curd itself should not simmer.)
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The curd will thicken and form a soft shape when dropped from a spoon.
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Stir in the zest after removing from the heat.
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Let cool to room temperature, scrape into an airtight container, and refrigerate overnight or for up to 1 week.
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For the frosting: put the coconut milk and sugar in a small saucepan.
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(Chances are that you will have a little over 1/2 cup of coconut milk left if you have used one 14-ounce can for the cake.
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Use it all to be efficient.
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It will be somewhere between 1/2 cup and 1/8 cup; thats okay.)
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Stir to combine, bring to a simmer over medium-high heat, and simmer until reduced by half.
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Cool completely.
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1) Put the heavy cream, reduced coconut milk, and confectioners sugar in a mixer bowl.
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Whip with the wire whip attachment until stiff peaks form, but do not overbeat.
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The frosting should be used immediately.
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For the assembly: toss together the 1 1/4 cups toasted coconut with the reserved 1 1/4 cups coconut in a large bowl.
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1) Place one cake layer on a 9-inch cardboard round, bottom side down.
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Spread the top of the cake layer with the lime cream; top with the second cake layer, bottom side up.
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Frost with the whipped cream frosting.
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Press the toasted coconut mixture all over the top and sides.
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Refrigerate for at least 4 hours or overnight.
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Bring to room temperature before serving.
