How to Make It
Preheat the oven to 350°. Butter two 8-inch round cake pans and line the bottoms with rounds of parchment or wax paper. Butter the paper and flour the pans, tapping out the excess.
In a large bowl, beat the egg yolks. Gradually add 1 cup of the sugar and beat until the mixture is as thick as mayonnaise. Beat in the poppy seeds and vanilla.
In another large bowl, using clean beaters, beat the egg whites at high speed for 6 seconds. Reduce the speed to moderately low and continue beating, gradually working your way back up to high speed. When soft peaks form, sprinkle in the remaining 3 tablespoons of sugar and beat to stiff shining peaks.
Scoop one-quarter of the whites into the yolk mixture and rapidly but delicately fold them in. Sift one-third of the flour over the batter and fold it in along with one-third of the egg whites. Continue to fold in one-third of the flour and one-third of the egg whites at a time until everything has been incorporated; fold in the melted butter with the last of the whites.
Immediately turn the batter into the prepared pans. Bang the pans once lightly on your work surface to force any air bubbles out of the batter. Bake in the lower third of the oven for 20 to 25 minutes; the cakes are done when they just begin to shrink from the pans. Let the cakes cool on a rack for 30 minutes. Invert the cakes, peel off the paper and let cool completely.
In a small saucepan, combine all of the ingredients and stir over low heat until the sugar is completely dissolved.
Whisk all of the ingredients in a medium stainless steel saucepan over very low heat until the bubbles begin to subside and a tiny wisp of steam appears; do not boil. Stir the curd over ice until cooled and lightly thickened.
In a medium bowl, beat the butter until softened, then beat in the butter until softened, then beat in the confectioners' sugar until light and fluffy. Beat in 1/3 cup of the cooled lemon curd until smooth.
Cut a 1/4-inch wedge out of the side of each cake; this will enable you to line up the layers when assembling the cake. Using a long serrated knife, slice each cake in half horizontally and set on a work surface, cut side up.
Gently brush each cake layer with 2 tablespoons of the rum syrup. Center a bottom layer of cake on a platter and spread one-third of the remaining lemon curd over it. Cover with the top layer, cut sides together, taking care to line up the wedges. Spread with lemon curd. Repeat with the remaining cake and lemon curd, ending with a cake layer.
Spread a very thin layer of the icing over the top and side of the cake to keep wandering crumbs in place. Then spread a 1/4-inch layer of icing over the top and side. Decorate as you wish, but I suggest using a pastry bag fitted with a star tip. In any case, sprinkling the cake with poppy seeds, as Child has here, can disguise certain ineptitudes.
Refrigerate the cake, uncovered, to firm up the icing. Then wrap the cake in plastic and refrigerate for up to 3 days or freeze for up to 1 month. Either way, be sure to allow the cake to come to room temperature before serving.