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1
Make cake:
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Put oven rack in middle position and preheat oven to 350u00b0F Butter and flour cake pans, knocking out excess flour.
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3
Sift together flour (2 3/4 cups), baking powder, and salt into a bowl.
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Beat together butter and sugar with an electric mixer (fitted with paddle attachment if using a stand mixer) at medium-high speed until pale and fluffy, 3 to 5 minutes. Beat in eggs 1 at a time, then beat in zest and vanilla and continue beating 5 minutes more. Reduce speed to low, then add flour mixture and milk alternately in 4 batches, beginning with flour mixture and mixing until batter is just smooth. Divide between cake pans, spreading evenly.
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5
Bake cake layers until they begin to pull away from sides of pans and a wooden pick or skewer comes out clean, 20 to 25 minutes. Cool 5 minutes in pans on racks, then invert cake layers onto racks and cool completely. (Leave oven on for toasting coconut.)
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6
Make filling while layers bake.
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7
Whisk together eggs in a heatproof bowl until combined well.
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8
With clean dry whisk, stir together sugar, cornstarch, and a pinch of salt in a 1 1/2- to 2-quart heavy saucepan, then whisk in water and juices until smooth. Bring to a boil over moderate heat, whisking, then reduce heat and cook at a bare simmer, whisking constantly, 2 minutes (mixture will be thick).
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9
Add half of hot juice mixture to eggs in a slow stream, whisking, then whisk egg mixture into juices in saucepan and cook over moderately low heat, whisking, just until it reaches a boil. Remove pan from heat, then add butter and zest, whisking until butter is melted. Chill filling, its surface covered with a buttered round of wax paper (buttered side down), until cold, about 30 minutes.
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10
Make frosting.
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11
Spread coconut in a shallow baking pan and toast in oven, stirring occasionally, until golden, 12 to 15 minutes.
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Beat together egg whites, sugar, water, corn syrup, and a pinch of salt in a large metal bowl with a handheld electric mixer (clean beaters if necessary) until combined. Set bowl over a saucepan of simmering water and beat mixture at high speed until it holds stiff, glossy peaks, 5 to 7 minutes. (Humid weather may necessitate additional beating time.) Remove bowl from heat, then add vanilla and lemon juice and continue beating until frosting is cooled and very thick, 6 to 10 minutes.
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13
Assemble cake:
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14
Halve each cake horizontally with a long serrated knife. Put 1 layer on a cake stand or large plate and spread with about 3/4 cup filling. Stack remaining cake layers using about 3/4 cup filling between each layer. Spread top and side of cake with frosting and coat cake with coconut, gently pressing to help it adhere.
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15
Cooks' notes: Cake layers (not split horizontally) can be made 3 days ahead and chilled, wrapped individually in plastic wrap. Filling can be chilled up to 8 hours. Stir before spreading. Frosting can be made 4 hours ahead and chilled, covered. Cake can be assembled and frosted 2 hours ahead.
