Preheat oven to 350 degrees. Process 2 1/2 ounces blackberries (12 to 15 berries), 1 tablespoon lemon juice, and 3 tablespoons sugar in a food processor or blender until smooth. Pass mixture through a fine sieve into a small bowl; discard solids. Set sauce aside.
Butter inner top inch of six 6-ounce ramekins; set aside. Whisk together egg yolks and 3/4 cup sugar in a medium bowl. Whisk in flour and milk in two batches each, beginning with the flour. Whisk in remaining 1/2 cup lemon juice, the salt, and the melted butter, and set aside.
Put egg whites in the bowl of an electric mixer fitted with the whisk attachment; beat on medium-high speed until very frothy, about 1 1/2 minutes. With mixer running, add remaining 1/4 cup sugar in a slow, steady stream; beat until whites hold stiff (but not dry) peaks, about 2 minutes. Whisk half the whites into reserved lemon mixture until combined; gently fold in remaining whites with a rubber spatula.
Place the ramekins in a high-sided roasting pan or baking dish, and divide the batter among ramekins, filling each almost to the top. Spoon a few drops of berry sauce onto cakes, and use a toothpick or skewer to swirl sauce into batter. Transfer to oven; pour boiling water into pan, a bit more than halfway up sides of ramekins.
Bake until cakes are set and the tops are just starting to turn golden brown, 35 to 40 minutes. With tongs, transfer the ramekins from pan to a wire rack, and let cool 15 minutes.
Meanwhile, beat creme fraiche in a clean mixing bowl until it holds soft peaks. Serve cakes warm with creme fraiche and remaining blackberries.