Measure 1 tbsp lemon juice into a small bowl; sprinkle gelatin over top. Heat remaining lemon juice, sugar, and salt in medium nonreactive saucepan over medium high heat, stirring occasionally, until sugar dissolves and mixture is not but not boiling. Whisk eggs and yolks in large nonreactive bowl. Whisking constantly, slowly pour hot lemon-sugar mixture into eggs, then return mixture to saucepan. Cook over medium-low heat, stirring constantly with heatproof spatula, until mixture registers 170 degrees on instant-read thermometer and is thick enough to leave a trail when spatula is scraped along pan bottom, 4 to 6 minutes. Immediately remove pan from heat and stir in gelatin mixture until dissolved. Stir in frozen butter until incorporated. Pour filling through fine-mesh strainer into nonreactive bowl (you should have 3 cups). Cover surface directly with plastic wrap; refrigerate until firm enough to spread, at least 4 hours.

Adjust oven rack to middle position and heat oven to 350 degrees. Grease and flour two 9-inch-wide by 2-inch-high round cake pans and line with parchment paper. In 2-cup liquid measure or medium bowl, whisk together milk, egg whites, and vanilla.

In bowl of standing mixer fitted with paddle attachment, mix flour, sugar, baking powder, and salt at low speed. With mixer running at low speed, add butter one piece at a time; continue beating until mixture resembles moist crumbs with no visible butter chunks. Add all but 1/2 cup milk mixture to crumbs and beat at medium speed until mixture is pale and fluffy, about 1 1/2 minutes. With mixture running at low speed, add remaining 1/2 cup milk mixture. Increase speed to medium and beat 30 seconds more. Stop mixer and scrape sides of bowl. Return mixer to medium speed and beat 20 seconds longer. Divide batter evenly between cake pans; using rubber spatu0la, spread batter to pan walls and smooth tops.

Bake until toothpick inserted in center of cakes comes out clean, 23-25 minutes (or about 1 hour 20 minutes for single cake pan). Loosen cakes from sides of pans with small knife, cool in pan 10 minutes, then invert onto greased wire rack; peel of parchment. Invert cakes again; cool completely on rack, about 1 1/2 hours.

Use a serrated knife to cut each cake into 2 even layers. Place bottom layer of 1 cake on cardboard round or cake plate. Using rubber spatula, spread 1 cup lemon filling evenly on cake, leaving a 1/4-inch border around edge; using cardboard round, gently replace top layer. Spread 1 cup filling on top. Using cardboard round, gently slide bottom half of second cake into place. Spread remaining cup filling on top. Using cardboard round, place top layer of second cake. Smooth out any filling that has leaked from sides of cake; cover with plastic wrap and refrigerate while making icing.

Combine all ingredients in bowl of standing mixer or large heatproof bowl and set over medium saucepan filled with 1 inch of barley simmering water (do not let bowl touch water). Cook, stirring constantly, until mixture registers 160 degrees on instant read thermometer, 5 to 10 minutes. Remove bowl from heat and transfer mixture to standing mixer fitted with whisk attachment. Beat on medium speed until soft peaks form, about 5 minutes. Increase speed to medium-high and continue to beat until mixture has cooled to room temperature and stiff beaks from, 5 minutes longer. Using icing spatula, spread frosting on cake.