Preheat the oven to 350°F.
Line the bottom of an 8 inch round cake pan with an 8 inch round piece of parchment paper and grease the sides and bottom to ensure easy release.
In a medium bowl, whisk together the all purpose flour, baking powder, and kosher salt until well combined.
In a large bowl mix or rub together the lemon zest and granulated sugar until combined and fragrant.
Using a handheld or stand mixer on medium speed, beat the butter and lemon sugar until light and fluffy, about 3-4 minutes. Add the eggs one at a time, beating after each until smooth.
Add the vanilla extract and lemon juice, beat until combined. Make sure to scrape down the bowl as needed, get the bottom really well.
In 2 batches, gradually add the dry ingredients to the butter mixture, alternating with the buttermilk until combined, do not over mix. Fold in the blackberries until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Position it on the center rack of the oven to ensure even baking, avoiding placement too close to the top or bottom. Bake for 28-32 minutes, or until a toothpick inserted into the center comes out clean and the top is golden. Try not to over bake, keep an eye on it!
When the cake is finished baking, let it cool in the pan for about 5 minutes. Then run a knife around the edges, place a plate on top, and carefully flip the cake out so the bottom becomes the top. This gives you a nice flat surface for the icing, and if any of the blackberries sank a bit while baking, they will end up on top.