Prepare the Curd: Place egg yolks in a small heatproof bowl. Bring sugar and lime juice to a boil in a heavy nonaluminum 3 1/2-quart saucepan over medium-high. Remove from heat. Gradually whisk about one fourth of the hot juice mixture into egg yolks. Add egg yolk mixture to remaining hot juice mixture in saucepan, whisking constantly until well combined. Cook mixture over medium, whisking constantly, until thickened to pudding-like consistency, 6 to 8 minutes. Add butter cubes in 6 additions, whisking constantly after each addition until butter melts and mixture is well combined. Remove from heat. Pour mixture through a fine mesh strainer into a bowl. Place plastic wrap directly on surface of mixture to prevent a film from forming. Chill at least 3 hours or up to 24 hours.

While Curd chills, prepare the Cake Layers: Preheat oven to 350°F. Grease (with shortening) and flour 3 (8-inch) round cake pans; set aside. Beat sugar and butter with a stand mixer fitted with a paddle attachment on medium speed until fluffy, 3 to 5 minutes. Gradually add egg whites, one third at a time, beating well on low speed after each addition. Whisk together flour, baking powder, and salt in a separate bowl. Gradually add flour mixture to butter mixture alternately with water, beginning and ending with flour mixture, beating on low speed until just incorporated. Stir in lime zest and vanilla. (Batter will appear broken.) Pour batter evenly into prepared cake pans. Bake in preheated oven until a wooden pick inserted in centers of cakes comes out clean, 18 to 21 minutes. Cool in pans on wire racks 10 minutes. Invert cakes onto wire racks, and let cool completely, about 1 hour.

Prepare the Frosting: Beat butter with an electric mixer on medium speed until smooth, about 1 minute. Gradually add powdered sugar, lime juice, vanilla, and salt, beating until smooth, about 2 minutes. Increase speed to medium-high; beat until light and fluffy, about 1 minute. Spoon Frosting into a large piping bag with a tip cut off (about 1/2 inch).

Assemble the Cake: Place 1 Cake Layer on a serving plate or a cake stand. Pipe Frosting in a ring (about 1/4-inch high) around top outer edge of layer. Spread one third of the Curd (about 1/2 cup) inside Frosting ring. Top with second Cake Layer; repeat Frosting piping and Curd spreading. Top with third Cake Layer; repeat Frosting piping and Curd spreading. Spread remaining Frosting around sides of cake, reserving about 1/2 cup in piping bag. Pipe reserved Frosting into a ring of 1/2-inch dots around top of cake. Garnish cake with lime zest.