Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch cake pan.
Mix both amounts of cake flour, baking powder, and salt in a bowl. Beat 3 egg yolks in a separate bowl with an electric mixer until thickened and lemon-colored, about 5 minutes.
Beat 2/3 cup white sugar, 3 1/2 tablespoons butter, and 3/4 teaspoon vanilla extract in a bowl with an electric mixer until light and fluffy; beat in egg yolks. Beat flour mixture in several additions, alternating with 6 tablespoons milk and ending with flour mixture, into creamed butter-sugar mixture on low until incorporated. Pour batter into the prepared cake pan.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, 20 to 25 minutes. Cool in the pan for 10 minutes before removing to cool completely on a wire rack.
Pour 1 1/2 cups milk into a saucepan; bring to a boil. Remove from heat. Whisk 4 egg yolks and 1/3 cup white sugar in a bowl until well combined; whisk in all-purpose flour. Pour 2 tablespoons of hot milk into egg mixture, whisking constantly. Slowly whisk in remaining hot milk. Return saucepan to stove and bring almost to a boil, stirring constantly, until thickened. Pour custard into a heat-proof bowl; stir in 1 teaspoon vanilla extract. Refrigerate until chilled.
Melt 3 tablespoons butter and chocolate in the top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Remove from heat; stir in confectioners' sugar and 3/4 teaspoon vanilla extract. Add water as needed to create desired consistency.
Cut cake in half horizontally. Place bottom cake layer on a serving platter. Spread cooled custard onto bottom half; place top cake over custard layer. Spread frosting on top and sides of cake. Chill until serving time.
