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1
Have all the ingredients at room temperature.
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2
Position a rack in the lower third of an oven and preheat to 350F Grease and flour three 8-inch round cake pans; tap out excess flour.
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3
To make the cake, over a sheet of waxed paper, sift together the flour, baking powder and salt; set aside.
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4
In a small bowl, stir together the milk and vanilla; set aside.
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5
In the bowl of an electric mixer fitted with the flat beater, beat the butter on medium speed until creamy and smooth, 1 to 2 minutes.
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6
Add the sugar and lime zest and continue beating until light and fluffy, about 5 minutes, stopping the mixer occasionally to scrape down the sides of the bowl.
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7
Add the eggs one at a time, beating well after each addition.
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8
Reduce the speed to low and add the flour mixture in three additions, alternating with the milk mixture and beginning and ending with the flour.
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9
Beat each addition just until incorporated, stopping the mixer occasionally to scrape down the sides of the bowl.
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10
Spoon the batter into the prepared pans, spreading the batter evenly.
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11
Bake until the cakes begin to pull away from the sides of the pans and a toothpick inserted into the center comes out clean, 25 to 30 minutes.
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12
Transfer the pans to a wire rack and let the cakes cool in the pans for 15 minutes, then turn them out onto the rack and let cool completely.
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13
Meanwhile, make the frosting: In the bowl of an electric mixer, using a handheld whisk, beat together the egg whites, sugar, corn syrup and salt just until combined, about 30 seconds.
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14
Place the bowl over but not touching simmering water in a saucepan and whisk until the sugar dissolves and the mixture is hot, 2 to 3 minutes.
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15
Set the bowl on the mixer fitted with the whisk attachment and beat on medium speed for 3 minutes.
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16
Add the vanilla and beat until the outside of the bowl is cool and medium-firm peaks form, 2 to 3 minutes.
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17
Using a rubber spatula, fold in 1 cup of the coconut just until incorporated.
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18
Place one cake layer, top side down, on a serving plate.
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19
Using an icing spatula or a knife, spread some of the frosting evenly on top.
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20
Place a second layer, top side down, on the first layer and spread some of the frosting evenly on top.
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21
Place the remaining layer, top side down, on the second layer.
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22
Spread the remaining frosting over the top and sides of the cake.
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23
Press the remaining 1 1/2 cups coconut onto the top and sides.
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24
Refrigerate the cake for 30 minutes before serving to set the frosting.