Make the cake: Heat oven to 350°. Butter and flour two 9″ cake pans, and set aside. Whisk together flour, baking powder, and salt in a bowl; set aside. Whisk together milk and vanilla in a bowl; set aside. In the bowl of a stand mixer fitted with a paddle, cream butter, 1 1⁄2 cups sugar, and zest on medium-high speed until pale and fluffy, about 3 minutes. Add eggs, one at a time, beating well after each addition. On low speed, alternately add dry ingredients in 3 batches and wet ingredients in 2 batches. Increase speed to high and beat until batter is smooth, about 5 seconds. Divide batter between prepared pans, and smooth top with a rubber spatula; drop pans lightly on a counter to expel any large air bubbles. Bake cakes until a toothpick inserted in middle comes out clean, about 30 minutes. Let cakes cool for 20 minutes in pans; invert onto wire racks, and let cool. Using a serrated knife, halve each cake horizontally to produce four layers; set aside. Bring remaining sugar and juice to a boil in a small saucepan over high heat. Remove from heat, and set syrup aside.
Make the frosting: Whisk together sugar, cornstarch, zest, and salt in a 4-qt. saucepan. Add yolks, and whisk until smooth; stir in juice. Stirring often, bring to a boil over medium heat; cook, stirring constantly, until very thick, about 3 minutes. Remove from heat, let cool, and transfer to a bowl; chill the lemon curd. In the bowl of a stand mixer fitted with a paddle, beat butter and 1⁄4 of the curd on medium-high speed until fluffy and smooth, about 1 minute. Add half the remaining curd, beating until smooth, and then add remaining curd and vanilla. Increase speed to high; beat frosting until pale and fluffy, about 3 minutes.
To assemble, place one cake layer on a cake stand, drizzle with 2 tbsp. syrup, and spread with 3⁄4 cup frosting; top with another cake, drizzle with 2 tbsp. syrup, and spread with 3⁄4 cup frosting. Place another cake over the frosting, drizzle with 2 tbsp. syrup, and spread with 3⁄4 cup frosting; top with remaining cake, and drizzle with remaining syrup. Cover top and sides with remaining frosting; chill cake to firm frosting. Serve at room temperature.