In a medium-sized saucepan, combine the blackberries, granulated sugar, and lemon juice. Cook over medium heat for about 10 minutes, stir occasionally until the mixture simmers. Strain to separate juices and seeds. Set aside blackberry purée.

Preheat your oven to 350°F (177°C) and grease a loaf pan.

Sift together the cake flour and salt in a bowl.

In a large mixing bowl, beat softened unsalted butter and granulated sugar with an electric mixer until pale and fluffy, about 5 minutes. Add vanilla extract and eggs one at a time.

Gradually add dry ingredients to the mixture, alternating with the sour cream, mixing until just combined.

Mix ½ cup of batter with the blackberry purée, then gently swirl this mixture into the remaining batter.

Pour batter into the prepared loaf pan and bake for 60-70 minutes, checking doneness with a toothpick.

Cool the cake in the pan for 20 minutes, then transfer to a wire rack. Whisk together blackberry juice, powdered sugar, milk, and vanilla extract for the icing and pour over the cooled cake.