For a more stable cake, you could cut the 9×13 cake in half the short way. But where’s the fun in that?
To 350°F (175°C). Line the bottom and sides of a 9×13-inch cake pan with parchment, and coat it lightly with nonstick cooking spray, as I find that meringue can stick even to parchment.
Make the meringue: In the large bowl of a stand mixer fitted with the whisk attachment, or with a hand mixer, beat the egg whites with 1/4 teaspoon salt on medium/low speed until they begin to thicken — they’ll look satiny and you’ll see some trails form from the beaters. Increase the speed to medium, and add half the sugar — 3/4 cup — a little at a time, letting each sprinkle disappear and beating 10 to 20 seconds before adding more. Add 1/2 teaspoon vanilla extract or paste and continue to whip the mixture until the egg whites are glossy, and stiff peaks form when the whisk is lifted. Set this bowl aside.
Make the cake: In a second bowl, but no need to clean your whisk/beaters if you’re using them again, beat the butter with the remaining 3/4 cup sugar, remaining 1/4 teaspoon salt, and lemon zest until light and fluffy. Add egg yolks and beat to combine. Sprinkle surface of batter with baking powder and beat thoroughly into mixture. Add the milk and beat to combine; the batter will immediately look curdly and split and like you’ve made a mistake. You haven’t. Add the flour and beat until smooth; the mixture should come back together evenly.
Bake the cake: Spoon the cake batter into the bottom of your prepared cake pan and spread it into a thin, even layer. Dollop the egg white mixture all over the batter and gently spread this across the cake. (For both, a small offset spatula makes this easier.) Bake the cake for 20 minutes, then check for doneness — a toothpick inserted into the cake should come out batter-free. This can be harder to assess under a meringue, so try a few places. The cake might need 5 to 10 minutes longer to set. [Yes, I found this long of a baking time range in testing.]
Once baked, cool cake in pan for 10 minutes, then used the parchment surrounding the cake to carefully yank the cake and parchment directly onto your cooling rack to completely cool.
Finish cake components: In a medium bowl, combine strawberries, 1 tablespoon of the sugar, and lemon juice and set aside. [It takes about 15 minutes for them to get juicy.] Combine the heavy cream, crème fraîche, remaining 1 tablespoon sugar, and vanilla extract or paste in a large, clean bowl, and beat mixture until soft peaks form.
Assemble the cake: Carefully cut the cooled cake in half the long way, creating two long rectangles. Loosen the parchment underneath, and carefully transfer the first half onto a plate. Swirl the top of it with half the whipped cream, then scratter with half the berries. [Will it be messy? Yes it will.] Place the second half of the cake on top of the berries and finish with remaining cream and berries. You can drizzle any extra juices from the bowl over the cake. Garnish with mint leaves and/or powdered sugar, if you wish.
To serve: Store the cake in the fridge until needed. Cut cake into messy, chaotic slices to serve. Leftovers keep in the fridge for 4 days.