Preheat the oven to 400°F (204°C). Line a 9" round pan with parchment paper and butter both the pan and the paper. Sift and whisk the flour, baking powder, baking soda, and salt into a bowl. Set the bowl aside.
Use an electric mixer to beat the butter and 2/3 cup of sugar until light and fluffy. Add the egg and vanilla extract. Turn the mixer to low speed and add the buttermilk and dry ingredients in 3 alternating batches, ending with the dry ingredients. Be careful not to overmix.
Turn off the mixer and use a spatula to gently fold in the dry ingredients until just blended. Pour the mixture into the prepared 9" pan and smooth the top with the spatula. Scatter the blackberries over the batter and lightly press them in. Sprinkle the remaining 1 1/2 tablespoons of sugar over the top.
Bake the cake for 30-35 minutes, or until a cake tester inserted in the center comes out clean. Remove the pan from the oven and let it sit for 10 minutes on a wire rack. Carefully turn out the cake, remove the parchment paper, and let it cool right side up on the wire rack. You can also serve it with whipped cream if desired.