Move an oven rack to the center position and preheat the oven to 325°F. Lightly grease an 8x8-inch baking dish and line it with parchment paper, leaving about 2 inches hanging over two opposite sides.
Separate the 4 eggs while they are still cold. Place yolks in a large bowl and whites in a very clean, dry mixing bowl. Let both bowls sit on the counter for 20 to 30 minutes to come to room temperature.
Melt the 4 tablespoons of butter and set it aside to cool slightly. Warm the 2 cups of milk in the microwave for 30 to 45 seconds, just until lukewarm and not hot.
Wash the lemons well, then zest 2 medium lemons until you have about 1 tablespoon. Add the 3/4 cup of granulated sugar to a small bowl and add the lemon zest. Rub the zest into the sugar with your fingertips for about 1 minute, until the sugar looks damp and smells strongly of lemon.
Juice the zested lemons and strain out the seeds, measuring 1/4 cup of fresh juice. Set the juice aside.
Add the lemon sugar to the bowl with the egg yolks. Beat with a mixer on medium-high speed for 2 to 3 minutes, until the mixture is pale yellow, thick, and slightly increased in volume.
With the mixer on low, slowly drizzle in the melted butter and mix until fully blended. Add the 1 teaspoon of vanilla extract and mix for a few seconds. Pour in the 1/4 cup of lemon juice and mix until smooth.
Add the 3/4 cup of flour and the 1/4 teaspoon of salt to the yolk mixture. Stir gently by hand with a whisk, just until no dry streaks remain. Pour in the lukewarm milk a little at a time, whisking constantly, until the batter is completely smooth.
Make sure your mixer bowl and whisk attachment are completely clean and dry. Add the egg whites and the 1/4 teaspoon of cream of tartar. Beat on medium speed for about 1 minute, until foamy, then increase to medium-high and whip for 3 to 4 minutes, until stiff peaks form.
Scoop about 1/2 cup of the whipped whites into the yolk batter and whisk it in fully. Add the remaining whites in three separate additions, using a large rubber spatula to fold gently. Stop when the batter looks like a frothy cloud with a few small pockets of white still floating on top.
Pour the batter slowly into the prepared baking dish. Smooth the top very gently with the spatula, then give the dish one light tap on the counter to pop any large bubbles.
Place the dish in the center of the oven and bake at 325°F for 40 to 50 minutes. Avoid opening the oven door during the first 30 minutes. The cake is done when the top is golden brown and the edges are set, but the center still has a slight jiggle.
Set the dish on a wire rack and let the cake cool completely in the pan, about 1 hour. Cover the dish with plastic wrap or foil and refrigerate for at least 2 hours, or overnight.
Hold the parchment overhang and lift the cold cake out of the dish onto a cutting board. Add the 2 tablespoons of powdered sugar to a fine-mesh sieve and tap it over the top for an even dusting. Slice into 9 squares with a sharp knife, wiping the blade clean between cuts.