Make the cake: Place a rack in center of oven, and preheat to 350 degrees. Grease and flour two deep 8-inch cake pans or springform pans, and set aside.
Place butter and sugar in a large mixing bowl and, with an electric mixer, beat on low speed until combined. Scrape down sides of bowl, increase mixer speed to medium, and beat 1 minute to lighten. Set aside.
Whisk together flour, baking powder, and salt in a medium bowl. Set aside.
Pour milk into a small bowl. Grate 2 teaspoons of lemon zest and add to bowl of milk. Cut lemon in half and squeeze 2 tablespoons juice, then add to milk. Stir and set aside.
Separate the eggs, placing whites in a large mixing bowl and yolks in a large, heavy saucepan. Set yolks aside. Beat whites with clean beaters on high speed until stiff peaks form, about 3 minutes. Set aside.
Spoon a third of flour mixture into creamed butter and sugar and beat on low speed 15 seconds. Add half of milk-lemon mixture and beat on low until just combined. Add another third of the flour, then the rest of milk mixture, and then the remaining flour mixture and beat until just combined. The batter will be thick. With a rubber spatula, fold in beaten egg whites until no whites remain, then divide batter between two pans. Place pans in oven.
Bake cakes until golden brown and the top springs back when lightly pressed with your finger, about 40 to 43 minutes. Remove pans from oven. Cool 15 minutes.
Make the filling and frosting: To the pan with the egg yolks, add sugar, flour, milk, and butter. Grate a generous tablespoon of lemon zest and squeeze ½ cup lemon juice; add both to pan. Stir to combine, and place pan over low heat. Cook and stir constantly until butter melts and mixture thickens, 4 to 5 minutes. When thickened, remove pan from heat and pour contents into a large mixing bowl to cool while preparing the cakes.
To assemble: Run a knife around edges of cake and invert onto a rack to cool completely, 20 minutes. Slice each cake in half horizontally. Place one layer on a cake plate and spread with a scant ¼ cup filling. Stack with another and repeat layers until you have enough filling and frosting to cover top and barely cover sides.
To finish, slice strawberries and pile on top of cake. This cake is best if left to rest for 1 to 2 hours before serving. If prepping ahead, garnish with berries just before serving.