Important: have your butter at a warm room temperature and your eggs at room temperature. For the eggs, you can place them in a bowl of hot water to warm them.
Grease and flour two 8 inch square (round works too) light colored metal pans. Preheat the oven to 350 F.
Into the bowl of a stand mixer, add the sugar. Zest the lemons over the sugar then use your fingers to rub the zest into the sugar to release the oils.
Add the butter, salt and vanilla and affix the paddle attachment. Turn the speed to medium and beat until creamed, about 3-5 minutes.
With the mixer on, and one by one, add the eggs. Allow 30 seconds between each egg addition. Once all three eggs are in, mix on medium speed until light and airy. (If the batter splits at this point, scroll above to read what to do).
Sift in the cake flour and add the baking powder. With the mixer on low, add the sour cream and fresh lemon juice. Mix until just combined - scraping down the bowl as needed to ensure an even mix.
Divide the batter between the two prepared pans, smooth the batter into an even layer. Arrange the blackberries on top, 16 berries per pan in rows of four (adding them on top means there will be slight sinking in the top of the cake, but adding them into the cake would have them all sinking to the bottom which we want to avoid).
Bake the cakes until the tops spring back when pressed in the middle (of the cake, not the berry), or a cake tester comes out clean, about 15-18 minutes.
Once the cakes cool slightly, carefully flip them over to release and transfer to a cooling rack to cool completely.