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TO MAKE THE CAKE: Preheat the oven to 350F.
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Line a 9 by 13-inch baking pan with parchment paper, grease with butter or cooking spray, dust the pan with flour, and knock out the excess.
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Using an electric mixer fitted with the paddle attachment, cream the butter and the 2 cups granulated sugar on medium-high speed until light and fluffy, about 3 minutes.
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Scrape down the sides of the bowl with a rubber spatula.
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Add the eggs, 1 at a time, beating after each addition.
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Beat until the batter is fluffy, an additional 2 minutes.
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In a bowl, stir together the 2 1/2 cups flour, baking powder, and salt.
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Stir the vanilla into the milk.
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Add the flour and milk mixtures in alternating batches, beginning and ending with flour.
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After each addition, mix on low speed just until the batter is smooth.
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Pour the batter into the prepared pan and bake until a toothpick inserted in the center comes out clean, 25 to 30 minutes.
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Remove from the oven and let cool for 10 minutes, then unmold on a wire rack to cool completely.
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TO MAKE THE FILLING: Stir together the 1 cup granulated sugar, 3 tablespoons flour, and 1/2 teaspoon salt in a saucepan.
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Gradually stir in the 1/4 cup water and the orange and lime juices.
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Cook over medium heat until the mixture boils.
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Remove from the heat and pour 1/4 cup of the hot mixture into a heatproof glass measuring cup.
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In a bowl, whisk together the egg yolks.
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Slowly pour the 1/4 cup hot juice mixture into the egg yolks, whisking constantly.
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(This tempers the egg yolks, preventing them from curdling.)
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In a slow, steady stream, pour the egg-juice mixture back into the saucepan with the remaining juice mixture, whisking constantly.
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Set the saucepan over medium heat and bring the mixture to a simmer, whisking constantly until it thickens, about 2 minutes.
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Pour the filling into a bowl, stir in the lime zest, and cover with plastic wrap, making sure the wrap directly touches the curd at all points, sealing out any air.
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Refrigerate until cool.
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TO MAKE THE FROSTING: Using an electric mixer fitted with the whisk attachment, beat the cream on high speed until soft peaks form.
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Whip in the powdered sugar.
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TO ASSEMBLE THE CAKE: Gently fold 1 cup of the whipped cream frosting into the cooled lime filling.
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Halve the cooled cake crosswise to make 2 layers.
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Place the bottom layer on a serving plate.
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Spread the filling over almost to the outer edge; top with the second layer and frost all over with the remaining whipped cream.
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Decorate with coconut and lime slices.
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Bake the cake layer in advance, wrap in plastic wrap and aluminum foil, and refrigerate or freeze for up to 3 weeks.
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The filling can be made up to 3 days in advance and refrigerated.
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Make the frosting and assemble the cake the day you plan to serve it.
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If it will be more than 2 hours before serving, refrigerate until ready to serve.
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To make decorative lime twists, slice a lime in 8 thin crosswise rounds; for each slice, cut through the skin, stopping in the middle of each round.
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Use your thumb and forefinger to grasp the lime on opposing sides of the cut and twist in opposite directions.
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Stand the lime circles, cut side down, on top of the cake.
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To toast the coconut: Preheat the oven to 350F.
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Place the coconut in a single layer on a rimmed baking sheet and bake until lightly golden around the edges, about 6 minutes.
