Place an oven rack in the center position and preheat oven to 350°F (175°C). Spray six 6-ounce ramekins generously with nonstick cooking spray or rub with softened butter, and set them inside a 9x13-inch baking dish. Put a kettle of water on to heat.
Wash and dry two large lemons. Zest until you have 1 tablespoon of fresh zest, avoiding the white pith. Cut and juice the lemons until you have 1/3 cup of fresh juice. Strain out any seeds and set zest and juice aside.
Crack each egg over a small bowl and pass the yolk back and forth between the shell halves so the white drips into the bowl. Place 3 yolks in a large mixing bowl and 3 whites in a second, completely clean and dry bowl. Make sure no yolk gets into the whites.
Whisk the egg yolks for about 30 seconds until smooth and slightly lighter. Add the milk, melted butter, lemon zest, lemon juice, and vanilla extract and whisk until combined. Add the flour, 1/2 cup of the sugar, and the salt, and whisk until completely smooth with no lumps.
Using an electric hand mixer, beat the egg whites on medium-low speed for 1 to 2 minutes until foamy. Increase speed to medium-high and beat until soft peaks form. Gradually add the remaining 1/4 cup of sugar, about a tablespoon at a time, while the mixer runs. Beat until glossy and holding soft to medium peaks, then stop.
Scoop about one-third of the whipped whites into the lemon batter and whisk gently to lighten it. Switch to a rubber spatula and add the remaining whites in two more additions. Fold by cutting down through the center, sweeping along the bottom, and lifting up and over. Stop when no large streaks remain.
Ladle the batter evenly into the six prepared ramekins, filling each nearly to the top. Carefully pull the oven rack out slightly and set the baking dish on it. Pour hot water from the kettle into the baking dish until it reaches about halfway up the sides of the ramekins, taking care not to splash water into the batter.
Slide the rack back in and bake for 30 to 35 minutes, until the tops look puffed, set, and lightly golden. An instant-read thermometer inserted into the center of one ramekin should read about 172 to 175°F. A gentle nudge should make the cake layer barely wiggle rather than slosh.
Carefully remove the baking dish from the oven and let the ramekins sit in the hot water for 10 minutes. Lift each ramekin out with tongs or a thick towel and transfer to a wire cooling rack. Cool for another 15 minutes for warm serving, or cool completely for chilled. Dust with powdered sugar and add fresh berries and a lemon slice before serving.